A New Family of Multipartition Graph Operations and Its Applications in Constructing Several Special Graphs

Abstract

A new family of graph operations based on multipartite graph with an arbitrary number of parts is defined and their applications are explored in this paper. The complete spectra of graphs derived from multipartite graphs are determined. Because the adjacency matrix of the multipartite graph is symmetric, we can use it to generate an unlimited number of special symmetric graphs. Methods for generating countless new families of integral graphs using these multipartite graph operations have been presented. By applying these multipartite graph operations, we can construct infinitely many orderenergetic graphs from orderenergetic or non-orderenergetic graphs. Additionally, infinite pairs of equienergetic and non-cospectral graphs can be generated through these new operations. Moreover, this kind of graph operation can also be used to construct other special graphs related to eigenvalues and energy.

1. Introduction

Let $G=\left(V\right(G),E(G\left)\right)$ be a graph with order $n,$ where $V\left(G\right)$ is the vertex set and $E\left(G\right)$ is the edge set. Usually, we use $K_{n_1,n_2,\dots ,n_k}$ and $K_n$ to denote the complete multipartite graph with order $n_1+n_2+\cdots +n_k$ and a complete graph of order $n.$ The m-splitting graph $Sp{l}_m\left(G\right)$ of a graph G is obtained by adding to each vertex v of G new m vertices, say $v_1,v_2,v_3,\dots ,v_m,$ such that $v_i,1\le i\le m$ is adjacent to each vertex that is adjacent to v in $G.$ The m-shadow graph $D_m\left(G\right)$ of a connected graph G is obtained by taking m copies of G, say $G_1,G_2,G_3,\dots ,G_m,$ then joining each vertex u in $G_i$ to the neighbors of the corresponding vertex v in $G_j,1\le i\le m,1\le j\le m.$

A duplicate graph of $G=(V,E)$ is $D\left(G\right)=(V_1,E_1),$ where the vertex set $V_1=V\bigcup W,$ and W is a set, such that $V\bigcap W=\varnothing ,\left|V\right|=\left|W\right|$ and $f:V⟶W$ is bijective (for $a\in V$, we write $f\left(a\right)$ as $a^{\prime }$ for convenience), the edge set $E_1$ of $D\left(G\right)$ is defined as the edge $ab$ is in E, if and only if both $a{b}^{\prime }$ and $a^{\prime }b$ are in $E_1.$ The adjacency matrix $A_G=\left(a_{ij}\right)$ of the graph G is a symmetric matrix, where $a_{ij}=1$ if $v_i{v}_j\in E\left(G\right),$ otherwise, $a_{ij}=0$. The characteristic polynomial of the graph G is the characteristic polynomial of the adjacency matrix $A_G$ and is defined as $f_G\left(\lambda \right).$ Let ${\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n$ be the eigenvalues of the adjacency matrix $A_G$; then, the spectrum of the graph G is $\left\{{\lambda }_1,{\lambda }_2,\dots ,{\lambda }_n\right\}.$ The energy $\mathcal{E}\left(G\right)$ of graph G is defined as

$$\mathcal{E}\left(G\right)=\sum _{i=1}^n\left|{\lambda }_i\right|.$$

This concept was first introduced by Gutman [1] as a way to approximate the total $\pi$-electron energy of a molecule, and has since been studied in a remarkably large number of papers [2,3,4,5,6].

The problem of classifying graphs based on their eigenvalues and energy has consistently received widespread attention in the academic community. A graph G is integral if all of its eigenvalues are integers [7]. Two graphs are said to be a pair of equienergetic and non-cospectral graphs if their energies are equal but their spectra are different [8]. The construction of the integral graphs and equienergetic graphs can be found in [9,10,11,12]. A graph G of order n is said to be borderenergetic if $\mathcal{E}\left(G\right)=2(n-1).$ This definition was first proposed in [13]. The borderenergetic graph has been studied in these papers [13,14,15,16,17]. If $\mathcal{E}\left(G\right)>2(n-1),$ then the graph is called a hyperenergetic graph [18] and if $\mathcal{E}\left(G\right)<n,$ the graph is called a hypoenergetic graph [19]. Recently, orderenergetic graphs were defined in [20]; a graph is said to be orderenergetic if its energy is equal to its order. These special graphs, related to eigenvalues and energy, have several applications in theoretical chemistry. It has been shown that complete bipartite graphs, $K_{p,p}$ and $K_{p,p,6p}$, are all connected orderenergetic graphs for any positive integer p, and computer searches have been conducted for all orderenergetic-connected graphs with orders of up to $10.$ In [21], the authors provided an infinite amount of complete multipartite orderenergetic graphs with an unusually large number of parts. Some researchers have also constructed orderenergetic graphs using the join of a regular graph, complete bipartite graphs and their complements [22,23,24]. The authors also generated orderenergetic, hypoenergetic, and equienergetic graphs by discussing the m-splitting, m-shadow, and duplicate graphs in [22].

In recent years, several new graph operations have been introduced to construct special graphs related to eigenvalues and energy in [25,26]. Based on this, we aim to explore an additional series of graph operations to generate these special graphs. In this paper, we propose a new family of graph operations based on non-complete multipartite graphs with an arbitrary number of parts. These non-complete multipartite graphs are integral regular graphs when the graph has an even number of parts. When the number of parts is odd, we also provide sufficient conditions for these multipartite graph operations to generate integral graphs. Through this new family of graph operations, we propose methods to generate an infinite amount of orderenergetic graphs and infinite pairs of equienergetic graphs. Compared with previous constructions, we did not use special graphs such as regular graphs, complete graphs, or complete multipartite graphs for our construction. Instead, we used non-complete multipartite graphs. Even though such multipartite graphs are normal when the graph has an even number of parts, the special graphs we generated differed, as the graph operations we employed were distinct.

This paper is organized as follows. In Section 2, we outline some known results that were needed to prove the main theorems. The new family of non-complete multipartite graph operations are introduced in Section 3. The spectra of the graphs derived from non-complete multipartite graph operations are expressed in Section 4. The applications of the non-complete multipartite graph operations are presented in Section 5. Finally, we conclude with the main results and discuss future works in Section 6.

2. Preliminaries

In this section, we recall the concepts of the floor and celling of a real number and list some known results from the m-splitting graph, as well as the theory of matrices as lemmas, which are needed to prove the main theorem [26].

The floor of a real number $x,$ written as $\lfloor x\rfloor ,$ is the largest integer that is less than or equal to $x.$ The celling of a real number $x,$ written as $\lceil x\rceil ,$ is the smallest integer that is greater than or equal to x [27]. The energy of the m-splitting graph $Sp{l}_m\left(G\right)$ can be calculated using the following result.

Lemma 1 

([22]). $\mathcal{E}\left(Sp{l}_m\left(G\right)\right)=\sqrt{1+4m}\mathcal{E}\left(G\right).$

Lemma 2 

([26]). If A and D are square matrices (need not be same order) and B and C are matrices with compatible orders, then the determinant of the following block matrix is given by

$$\begin{array}{cc}\hfill Det\left[\begin{array}{cc}A& B\\ C& D\end{array}\right]& =Det\left(D\right)Det(A-B{D}^{-1}C)=Det\left(A\right)Det(D-C{A}^{-1}B)\hfill \end{array}$$

(1)

provided $D^{-1}$ or $A^{-1}$ exists.

Lemma 3 

([25]). Let A be an $r^{th}$-order square matrix with eigenvalues $\left\{{\alpha }_i\right\},1\le i\le r$ and B be an $s^{th}$ order square matrix with eigenvalues $\left\{{\beta }_i\right\},1\le i\le s.$ Then, the eigenvalues of the square matrix $A\otimes B$ of the order $rs$ are given by all possible products $\left\{{\alpha }_i{\beta }_j\right\},$ for $1\le i\le r$ and $1\le j\le s,$ which is $rs$ in number.

Lemma 4 

([26]). If $B=\left[b_{ij}\right]$ is a $n^{th}$ order square matrix, then

$$J_{mn}B{J}_{nm}=\left(\sum _{i=1}^n\sum _{j=1}^m{b}_{ij}\right)J_{mm}.$$

3. Multipartite Graph Operations

In this section, the new multipartite graph operations will be proposed as follows. Let G be a graph with order n and number of edges $e.$ Take p copies of $G,$ denoted by $G_i$ and take $n\lfloor {\displaystyle \frac{m}{2}}\rfloor q,n\left(\lceil {\displaystyle \frac{m}{2}}\rceil -1\right)p$ number of isolated vertices for some positive integer $q,m,p.$ Let the n vertices of $G_i$ be $V\left(G_i\right)=\left\{v_{i1},v_{i2},\cdots ,v_{in}\right\},$ for $1\le i\le p.$ Let the isolated vertices be named as $V\left(U_j^{\left(r\right)}\right)=\left\{u_{jk}^{\left(r\right)}\right\}$ and $V\left(W_o^{\left(s\right)}\right)=\left\{w_{ot}^{\left(s\right)}\right\},$ where $1\le j\le q,1\le r\le \lfloor {\displaystyle \frac{m}{2}}\rfloor ,1\le o\le p,1\le s\le \lceil {\displaystyle \frac{m}{2}}\rceil -1$ and $1\le k,t\le n.$ Then construct the new graph $H_G^{(m,p,q)}=(V\left(H_G^{(m,p,q)}\right),E\left(H_G^{(m,p,q)}\right))$ from the graphs $G_i$ and the isolated vertices $\left\{u_{jk}^{\left(r\right)}\right\},\left\{w_{ot}^{\left(s\right)}\right\}$ as follows.

• $V\left(H_G^{(m,p,q)}\right)=\left({\bigcup }_{i=1}^{p}V\left(G_i\right)\right)\bigcup \left({\bigcup }_{r=1}^{\lfloor {\displaystyle \frac{m}{2}}\rfloor }\left({\bigcup }_{j=1}^{q}V\left(U_j^{\left(r\right)}\right)\right)\right)\bigcup \left({\bigcup }_{s=1}^{\lceil {\displaystyle \frac{m}{2}}\rceil -1}\left({\bigcup }_{o=1}^{p}V\left(W_o^{\left(s\right)}\right)\right)\right).$
• The edges in $H_G^{(m,p,q)}$ are obtained as follows. If $v_{ij}$ is adjacent to $v_{ik}$ in $G_i$ for $1\le i\le p,$ then
  • the edges $\left(v_{ij},u_{lk}^{\left(r\right)}\right)\in E\left(H_G^{(m,p,q)}\right)$ and $\left(v_{ik},u_{lj}^{\left(r\right)}\right)\in E\left(H_G^{(m,p,q)}\right)$ for all $1\le i\le p,1\le l\le q,1\le r\le \lfloor {\displaystyle \frac{m}{2}}\rfloor .$
  • the edges $\left(v_{ij},w_{ik}^{\left(s\right)}\right)\in E\left(H_G^{(m,p,q)}\right)$ and $\left(v_{ik},w_{ij}^{\left(s\right)}\right)\in E\left(H_G^{(m,p,q)}\right)$ for all $1\le i\le p,1\le s\le \lceil {\displaystyle \frac{m}{2}}\rceil -1.$
  • the edges $\left(u_{ij}^{\left(r\right)},w_{lk}^{\left(s\right)}\right)\in E\left(H_G^{(m,p,q)}\right)$ and $\left(u_{ik}^{\left(r\right)},w_{lj}^{\left(s\right)}\right)\in E\left(H_G^{(m,p,q)}\right)$ for all $1\le i\le q$ and $1\le l\le p,1\le r\le \lfloor {\displaystyle \frac{m}{2}}\rfloor ,1\le s\le \lceil {\displaystyle \frac{m}{2}}\rceil -1.$
  • the edges $\left(u_{ij}^{\left(r\right)},u_{ik}^{\left(r^{\prime }\right)}\right)\in E\left(H_G^{(m,p,q)}\right)$ and $\left(u_{ik}^{\left(r\right)},u_{ij}^{\left(r^{\prime }\right)}\right)\in E\left(H_G^{(m,p,q)}\right)$ for all $1\le i\le q,1\le r,r^{\prime }\le \lfloor {\displaystyle \frac{m}{2}}\rfloor ,$ and $r\ne r^{\prime }.$
  • the edges $\left(w_{ij}^{\left(s\right)},w_{ik}^{\left(s^{\prime }\right)}\right)\in E\left(H_G^{(m,p,q)}\right)$ and $\left(w_{ik}^{\left(s\right)},w_{ij}^{\left(s^{\prime }\right)}\right)\in E\left(H_G^{(m,p,q)}\right)$ for all $1\le i\le p,1\le s,s^{\prime }\le \lceil {\displaystyle \frac{m}{2}}\rceil -1,$ and $s\ne s^{\prime }.$

From the construction of these edges, it is easy to see that the new graph $H_G^{(m,p,q)}=(V\left(H_G^{(m,p,q)}\right),E\left(H_G^{(m,p,q)}\right))$ is a symmetric graph. The total number of vertices in the derived graph $H_G^{(m,p,q)}$ will be $n(\lceil {\displaystyle \frac{m}{2}}\rceil p+\lfloor {\displaystyle \frac{m}{2}}\rfloor q)$ and the total number of edges will be $e(2\lfloor {\displaystyle \frac{m}{2}}\rfloor \lceil {\displaystyle \frac{m}{2}}\rceil pq+\lceil {\displaystyle \frac{m}{2}}\rceil (m-2)p).$

It is not hard to see that the adjacency matrix of the new graph $H_G^{(m,p,q)}$ derived from the defined operations is given by the following Kronecker product

$$A_{H_G^{(m,p,q)}}=A_{M^{(m,p,q)}}\otimes A_G,$$

where $A_G$ is the adjacency matrix of the graph $G.$ $A_{M^{(m,p,q)}}$ is a square matrix of order $m,$ whose block matrix representation is given by

$$A_{M^{(m,p,q)}}={\left[\begin{array}{cccccc}{0}_p& J_{pq}& I_p& J_{pq}& I_p& \cdots \\ J_{qp}& 0_q& J_{qp}& I_q& J_{qp}& \cdots \\ I_p& J_{pq}& 0_p& J_{pq}& I_p& \cdots \\ J_{qp}& I_q& J_{qp}& 0_q& J_{qp}& \cdots \\ I_p& J_{pq}& I_p& J_{pq}& 0_p& \cdots \\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\end{array}\right]}_{m\times m},$$

where $J_{qp}$ represents the all one matrix of order $p\times q,$ $I_p$ represents the identity matrix of order p and $0_p$ represents a zero matrix of order $p.$ It is clear that this block matrix $A_{M^{(m,p,q)}}$ corresponds exactly to the adjacency matrix of a multipartite graph, which we will denote as $M^{(m,p,q)}.$

From the adjacency matrix of the multipartite graph $M^{(m,p,q)},$ it can be seen that the degree of the vertices of $M^{(m,p,q)}$ falls into two categories, $⌊{\displaystyle \frac{m}{2}}⌋q+⌈{\displaystyle \frac{m}{2}}⌉-1$ and $⌈{\displaystyle \frac{m}{2}}⌉p+⌊{\displaystyle \frac{m}{2}}⌋-1.$ Then the graph $M^{(m,p,q)}$ is regular when m is even and $p=q,$ or when $m=2k+1$ is odd and $(k+1)p=kq+1.$ However, $M^{(m,p,q)}$ will not be a complete graph because its adjacency matrix contains the identity matrix.

When $m=3,$ we have the following example.

Example 1. 

Let G be a graph with order n and number of edges $e.$ Take p copies of $G,$ denoted by $G_i,$ for $1\le i\le p,$ and take $nq,np$ number of isolated vertices for some positive integer $q,p.$ Let the n vertices of $G_i$ be $V\left(G_i\right)=\left\{v_{i1},v_{i2},\cdots ,v_{in}\right\},$ for $1\le i\le p.$ Let the isolated vertices be named as $V\left(U_j\right)=\left\{u_{jk}\right\}$ and $V\left(W_s\right)=\left\{w_{st}\right\},$ where $1\le j\le q,1\le s\le p$ and $1\le k,t\le n.$ A new graph $H_G^{(3,p,q)}=(V\left(H_G^{(3,p,q)}\right),E\left(H_G^{(3,p,q)}\right))$ can be constructed by the graph $G_i$ and the isolated vertices $\left\{u_{jk}\right\}$ and $\left\{w_{st}\right\}$ as follows.

1. 

$V\left(H_G^{(3,p,q)}\right)=\left({\bigcup }_{i=1}^{p}V\left(G_i\right)\right)\bigcup \left({\bigcup }_{j=1}^{q}V\left(U_j\right)\right)\bigcup \left({\bigcup }_{s=1}^{p}V\left(W_s\right)\right).$

2. 

The edges in $H_G^{(3,p,q)}$ are obtained as follows. If $v_{ij}$ is adjacent to $v_{ik}$ in $G_i$ for $1\le i\le p,$ then

• the edges $\left(v_{ij},u_{lk}\right)\in E\left(H_G^{(3,p,q)}\right)$ and $\left(v_{ik},u_{lj}\right)\in E\left(H_G^{(3,p,q)}\right)$ for all $1\le i\le p,1\le l\le q.$
• the edges $\left(v_{ij},w_{ik}\right)\in E\left(H_G^{(3,p,q)}\right)$ and $\left(v_{ik},w_{ij}\right)\in E\left(H_G^{(3,p,q)}\right)$ for all $1\le i\le p.$
• the edges $\left(u_{ij},w_{lk}\right)\in E\left(H_G^{(3,p,q)}\right)$ and $\left(u_{ik},w_{lj}\right)\in E\left(H_G^{(3,p,q)}\right)$ for all $1\le i\le q$ and $1\le l\le p.$

The total number of vertices in the derived graph $H^{(3,p,q)}$ will be $n(2p+q)$ and the total number of edges will be $2e(2pq+p).$

It is not hard to see that the block matrix $A_{M^{(3,p,q)}}$ corresponding to this new graph $H_G^{(3,p,q)}$ can be expressed as follows.

$$A_{M^{(3,p,q)}}=\left[\begin{array}{ccc}{0}_p& J_{pq}& I_p\\ J_{qp}& 0_q& J_{qp}\\ I_p& J_{pq}& 0_p\end{array}\right]$$

The graph $H_{K_2}^{(3,2,3)}$ is derived from the complete graph $K_2$ and the graph operation based on multipartite graph $M^{(3,2,3)}$ that is shown in Figure 1.

When the order of the multipartite graph is even, we set $p=q$ and use $M^{(m,p)}$ to replace $M^{(m,p,p)}.$ The following multipartite graph operation is from $m=4.$

Example 2. 

Let G be a graph with order n and number of edges $e.$ Take p copies of $G,$ denoted by $G_i,$ for $1\le i\le p,$ and take $3np$ number of isolated vertices for positive integer $p.$ Let the n vertices of $G_i$ be $V\left(G_i\right)=\left\{v_{i1},v_{i2},\cdots ,v_{in}\right\},$ for $1\le i\le p.$ Let the isolated vertices be named as $V\left(U_j\right)=\left\{u_{jk}\right\},$ $V\left(W_s\right)=\left\{w_{st}\right\}$ and $V\left(Y_o\right)=\left\{y_{od}\right\},$ where $1\le j,s,o\le p,$ and $1\le k,t,d\le n.$ The new graph $H_G^{(4,p)}=(V\left(H_G^{(4,p)}\right),E\left(H_G^{(4,p)}\right))$ can be constructed by the graph $G_i$ and the isolated vertices $\left\{u_{jk}\right\},$ $\left\{w_{st}\right\}$ and $\left\{y_{od}\right\}$ as follows.

1. 

$V\left(H_G^{(4,p)}\right)=\left({\bigcup }_{i=1}^{p}V\left(G_i\right)\right)\bigcup \left({\bigcup }_{j=1}^{p}V\left(U_j\right)\right)\bigcup \left({\bigcup }_{s=1}^{p}V\left(W_s\right)\right)\bigcup$

$\left({\bigcup }_{o=1}^{p}V\left(Y_o\right)\right).$

2. 

The edges in $H_G^{(4,p)}$ are obtained as follows. If $v_{ij}$ is adjacent to $v_{ik}$ in $G_i$ for $1\le i\le p,$ then

• the edges $\left(v_{ij},u_{lk}\right)\in E\left(H_G^{(4,p)}\right),$ $\left(v_{ik},u_{lj}\right)\in E\left(H_G^{(4,p)}\right),$ $\left(v_{ij},y_{ok}\right)\in E\left(H_G^{(4,p)}\right)$ and $\left(v_{ik},y_{oj}\right)\in E\left(H_G^{(4,p)}\right)$ for all $1\le i,l,o\le p.$
• the edges $\left(v_{ij},w_{ik}\right)\in E\left(H_G^{(4,p)}\right)$ and $\left(v_{ik},w_{ij}\right)\in E\left(H_G^{(4,p)}\right)$ for all $1\le i\le p.$
• the edges $\left(u_{ij},w_{lk}\right)\in E\left(H_G^{(4,p)}\right)$ and $\left(u_{ik},w_{lj}\right)\in E\left(H_G^{(4,p)}\right)$ for all $1\le i,l\le p.$
• the edges $\left(u_{ij},y_{ik}\right)\in E\left(H_G^{(4,p)}\right)$ and $\left(y_{ik},u_{ij}\right)\in E\left(H_G^{(4,p)}\right)$ for all $1\le i\le p.$
• the edges $\left(w_{ij},y_{lk}\right)\in E\left(H_G^{(4,p)}\right)$ and $\left(w_{ik},y_{lj}\right)\in E\left(H_G^{(4,p)}\right)$ for all $1\le i,l\le p.$

The total number of vertices in the derived graph $H_G^{(4,p)}$ will be $4np$ and the total number of edges will be $2e(4p^2+2p).$

It is not hard to see that the block matrix $A_{M^{(4,p)}}$ corresponding to this new graph $H_G^{(4,p)}$ can be expressed as follows.

$$A_{M^{(4,p)}}=\left[\begin{array}{cccc}{0}_p& J_p& I_p& J_p\\ J_p& 0_p& J_p& I_p\\ I_p& J_p& 0_p& J_p\\ J_p& I_p& J_p& 0_p\end{array}\right]$$

The graph $H_{K_2}^{(4,2)}$ is derived from the complete graph $K_2$ and the graph operation based on multipartite graph $M^{(4,2)}$ that is shown in Figure 2.

4. Spectra of the Graphs Derived from the Multipartite Graph Operations

The spectra of the graphs $H_G^{(2k+1,p,q)}$ and $H_G^{(2k,p)}$ will be found out in this section respectively, where k is any positive integer. The graph $H_G^{(2k+1,p,q)}$ is derived from a graph G and the operation based on multipartite graph $M^{(2k+1,p,q)}$ with odd order. The graph $H_G^{(2k,p)}$ is derived from a graph G and the operation based on multipartite graph $M^{(2k,p)}$ with even order.

4.1. Spectrum of the Graph $H_G^{(2k+1,p,q)}$

Firstly, we calculate the complete spectrum of the graph $H_G^{(2k+1,p,q)}.$ The spectrum of Example 1 was then proposed.

Theorem 1. 

If G is a graph of order n whose non-zero eigenvalues are given by $\left\{{\lambda }_i\right\},1\le i\le r,$ for some $r\le n,$ then the non-zero eigenvalues of the graph $H_G^{(2k+1,p,q)}$ derived from G and $M^{(2k+1,p,q)}$ are $\left\{-{\lambda }_i\right\},1\le i\le r,$ with multiplicity $pk+q(k-1),$ $\left\{(k-1){\lambda }_i\right\},1\le i\le r,$ with multiplicity $q-1,$ and $\left\{k{\lambda }_i\right\},1\le i\le r,$ with multiplicity $p-1,$ $\left\{{\displaystyle \frac{(2k-1)\pm \sqrt{4k^{2}pq+4kpq+1}}{2}}{\lambda }_i\right\}$ for $1\le i\le r,$ which are $2r$ numbers. Also, zero is an eigenvalue $H_G^{(2k+1,p,q)}$ with multiplicity $\left[(k+1)p+kq\right](n-r).$

Proof. 

Since the characteristic polynomial of the adjacency matrix of multipartite graph $M^{(2k+1,p,q)}$ is given by

$$\begin{array}{cc}\hfill f_{M^{(2k+1,p,q)}}\left(\lambda \right)& =Det\left(\left[\begin{array}{cccccc}\lambda I_p& -J_{pq}& -I_p& -J_{pq}& \cdots & -I_p\\ -J_{qp}& \lambda I_q& -J_{qp}& -I_q& \cdots & -J_{qp}\\ -I_p& -J_{pq}& \lambda I_p& -J_{pq}& \cdots & -I_p\\ -J_{qp}& -I_q& -J_{qp}& \lambda I_q& \cdots & -J_{qp}\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ -I_p& -J_{pq}& -I_p& -J_{pq}& \cdots & \lambda I_p\end{array}\right]\right).\hfill \end{array}$$

According to Lemma 2,

$$\begin{array}{cc}\hfill & f_{M^{(2k+1,p,q)}}\left(\lambda \right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{cccccc}\lambda I_p& -J_{pq}& -I_p& -J_{pq}& \cdots & -J_{pq}\\ -J_{qp}& \lambda I_q& -J_{qp}& -I_q& \cdots & -I_q\\ -I_p& -J_{pq}& \lambda I_p& -J_{pq}& \cdots & -J_{pq}\\ -J_{qp}& -I_q& -J_{qp}& \lambda I_q& \cdots & -I_q\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ -J_{qp}& -I_q& -J_{qp}& -I_q& \cdots & \lambda I_q\end{array}\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{\lambda }}\left[\begin{array}{c}-I_p\\ -J_{qp}\\ -I_p\\ -J_{qp}\\ ⋮\\ -J_{qp}\end{array}\right]\left[\begin{array}{cccccc}-I_p& -J_{pq}& -I_p& -J_{pq}& \cdots & -J_{pq}\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{cccc}(\lambda -{\displaystyle \frac{1}{\lambda }})I_p& -(1+{\displaystyle \frac{1}{\lambda }})J_{pq}& \cdots & -(1+{\displaystyle \frac{1}{\lambda }})J_{pq}\\ -(1+{\displaystyle \frac{1}{\lambda }})J_{qp}& \lambda I_q-{\displaystyle \frac{p}{\lambda }}{J}_q& \cdots & -I_q-{\displaystyle \frac{p}{\lambda }}{J}_q\\ -(1+{\displaystyle \frac{1}{\lambda }})I_p& -(1+{\displaystyle \frac{1}{\lambda }})J_{pq}& \cdots & -(1+{\displaystyle \frac{1}{\lambda }})J_{pq}\\ -(1+{\displaystyle \frac{1}{\lambda }})J_{qp}& -I_q-{\displaystyle \frac{p}{\lambda }}{J}_q& \cdots & -I_q-{\displaystyle \frac{p}{\lambda }}{J}_q\\ ⋮& ⋮& ⋮& ⋮\\ -(1+{\displaystyle \frac{1}{\lambda }})J_{qp}& -I_q-{\displaystyle \frac{p}{\lambda }}{J}_q& \cdots & \lambda I_q-{\displaystyle \frac{p}{\lambda }}{J}_q\end{array}\right]\right)\hfill \end{array}$$

Subtract the first row from all odd rows and the second row from all even rows.

$$\begin{array}{cc}\hfill & f_{M^{(2k+1,p,q)}}\left(\lambda \right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{cccc}(\lambda -{\displaystyle \frac{1}{\lambda }})I_p& -(1+{\displaystyle \frac{1}{\lambda }})J_{pq}& \cdots & -(1+{\displaystyle \frac{1}{\lambda }})J_{pq}\\ -(1+{\displaystyle \frac{1}{\lambda }})J_{qp}& \lambda I_q-{\displaystyle \frac{p}{\lambda }}{J}_q& \cdots & -I_q-{\displaystyle \frac{p}{\lambda }}{J}_q\\ -(1+\lambda )I_p& 0& \cdots & 0\\ 0& -(1+\lambda )I_p& \cdots & 0\\ -(1+\lambda )I_p& 0& \cdots & 0\\ ⋮& ⋮& ⋮& ⋮\\ -(1+\lambda )I_p& 0& \cdots & 0\\ 0& -(1+\lambda )I_p& \cdots & (1+\lambda )I_p\end{array}\right]\right)\hfill \end{array}$$

We add all odd columns to the first column and all even columns to the second column, then apply the Lemma 2 to obtain the following result.

$$\begin{array}{cc}\hfill & f_{M^{(2k+1,p,q)}}\left(\lambda \right)\hfill \\ \hfill & ={\lambda }^p{(\lambda +1)}^{(p+q)(k-1)}Det\left(\left[\begin{array}{cc}(\lambda -{\displaystyle \frac{k+(k-1)\lambda }{\lambda }})I_p& -{\displaystyle \frac{k(1+\lambda )}{\lambda }}{J}_{pq}\\ -{\displaystyle \frac{k(1+\lambda )}{\lambda }}{J}_{qp}& (\lambda -k+1)I_q-{\displaystyle \frac{kp}{\lambda }}{J}_q\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^p{(\lambda +1)}^{(p+q)(k-1)}{\left(\lambda -{\displaystyle \frac{k}{\lambda }}-(k-1)\right)}^{p}Det\left((\lambda -(k-1))I_q-{\displaystyle \frac{kp}{\lambda }}{J}_q\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{k^{2}p{(\lambda +1)}^2}{\lambda ({\lambda }^2-(k-1)\lambda -k)}}{J}_q\right)\hfill \\ \hfill & ={(\lambda +1)}^{(p+q)(k-1)}{(\lambda +1)}^p{(\lambda -k)}^{p-1}{(\lambda -(k-1))}^{q-1}{\displaystyle \frac{1}{\lambda }}\left[\lambda (\lambda -(k-1\left)\right)\right.\hfill \\ \hfill & \left.(\lambda -k)-kqp(\lambda -k)-k^{2}qp{(\lambda +1)}^2\right]\hfill \\ \hfill & ={(\lambda +1)}^{pk+q(k-1)}{(\lambda -k)}^{p-1}{(\lambda -(k-1))}^{q-1}\left({\lambda }^2-(2k-1)\lambda +k^2-k\right.\hfill \\ \hfill & \left.-kqp-k^{2}qp\right)\hfill \end{array}$$

So, the non-zero eigenvalues of the matrix $A_{M^{(2k+1,p,q)}}$ are $-1$ with multiplicity $pk+q(k-1),$ $k-1$ with multiplicity $q-1,$ k with multiplicity $p-1,{\displaystyle \frac{(2k-1)\pm \sqrt{4k^{2}pq+4kpq+1}}{2}}.$ Let $A_G$ and $A_{H_G^{(2k+1,p,q)}}$ are the adjacency matrices of the graphs G and $H_G^{(2k+1,p,q)}$ respectively, then $A_{H_G^{(2k+1,p,q)}}=A_{M^{(2k+1,p,q)}}\otimes A_G$. Then from Lemma 3 the theorem follows. □

When $k=1,$ the following corollary can be obtained.

Corollary 1. 

If G is a graph of order n whose non-zero eigenvalues are given by $\left\{{\lambda }_i\right\},1\le i\le r,$ for some $r\le n,$ then the non-zero eigenvalues of the graph $H_G^{(3,p,q)}$ derived from G and $M^{(3,p,q)}$ are $\left\{-{\lambda }_i\right\},1\le i\le r,$ with multiplicity $p,$ $\left\{{\lambda }_i\right\},1\le i\le r,$ with multiplicity $p-1,$ $\left\{{\displaystyle \frac{1\pm \sqrt{1+8qp}}{2}}{\lambda }_i\right\}$ for $1\le i\le r,$ which are $2r$ number. Also, zero is an eigenvalue $H_G^{(3,p,q)}$ with multiplicity $(2p+q)n-(2p+1)r.$

Proof. 

Since the characteristic polynomial of the adjacency matrix of multipartite graph $M^{(3,p,q)}$ is given by

$$\begin{array}{cc}\hfill f_{M^{(3,p,q)}}\left(\lambda \right)& =Det\left(\left[\begin{array}{ccc}\lambda I_p& -J_{pq}& -I_p\\ -J_{qp}& \lambda I_q& -J_{qp}\\ -I_p& -J_{pq}& \lambda I_p\end{array}\right]\right).\hfill \end{array}$$

According to Lemma 2,

$$\begin{array}{cc}\hfill & Det\left(\left[\begin{array}{ccc}\lambda I_p& -J_{pq}& -I_p\\ -J_{qp}& \lambda I_q& -J_{qp}\\ -I_p& -J_{pq}& \lambda I_p\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{cc}\lambda I_p& -J_{pq}\\ -J_{qp}& \lambda I_q\end{array}\right]-\left[\begin{array}{c}-I_p\\ -J_{qp}\end{array}\right]{\displaystyle \frac{I_p}{\lambda }}\left[\begin{array}{cc}-I_p& -J_{pq}\end{array}\right]\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{cc}\lambda I_p& -J_{pq}\\ -J_{qp}& \lambda I_q\end{array}\right]-{\displaystyle \frac{1}{\lambda }}\left[\begin{array}{c}-I_p\\ -J_{qp}\end{array}\right]\left[\begin{array}{cc}-I_p& -J_{pq}\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{cc}\lambda I_p& -J_{pq}\\ -J_{qp}& \lambda I_q\end{array}\right]-{\displaystyle \frac{1}{\lambda }}\left[\begin{array}{cc}{I}_p& J_{pq}\\ J_{qp}& p{J}_q\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{cc}\left(\lambda -{\displaystyle \frac{1}{\lambda }}\right)I_p& (-1-{\displaystyle \frac{1}{\lambda }})J_{pq}\\ (-1-{\displaystyle \frac{1}{\lambda }})J_{qp}& \lambda I_q-{\displaystyle \frac{p}{\lambda }}{J}_q\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^p{\left(\lambda -{\displaystyle \frac{1}{\lambda }}\right)}^{p}Det\left(\lambda I_q-{\displaystyle \frac{p}{\lambda }}{J}_q-{\displaystyle \frac{\lambda }{{\lambda }^2-1}}{\displaystyle \frac{p{(\lambda +1)}^2}{{\lambda }^2}}{J}_q\right)\hfill \\ \hfill & ={\left({\lambda }^2-1\right)}^{p}Det\left(\lambda I_q-{\displaystyle \frac{p}{\lambda }}{J}_q-{\displaystyle \frac{p(\lambda +1)}{\lambda (\lambda -1)}}{J}_q\right)\hfill \\ \hfill & ={\lambda }^{q-1}{\left(\lambda +1\right)}^p{\left(\lambda -1\right)}^{p-1}\left({\lambda }^2-\lambda -2pq\right).\hfill \end{array}$$

So, the non-zero eigenvalues of the matrix $A_{M^{(3,p,q)}}$ are $-1$ with multiplicity $p,$ 1 with multiplicity $p-1,{\displaystyle \frac{1\pm \sqrt{1+8qp}}{2}}.$ Let $A_G$ and $A_{H_G^{(3,p,q)}}$ are the adjacency matrices of the graphs G and $H_G^{(3,p,q)}$ respectively, then $A_{H_G^{(3,p,q)}}=A_{M^{(3,p,q)}}\otimes A_G$. Then from Lemma 3 the corollary follows. □

4.2. Spectrum of the Graph $H_G^{(2k,p)}$

Firstly, we calculate the complete spectrum of the graph $H_G^{(2k,p)}.$ The spectrum of Example 2 was then given.

Theorem 2. 

If G is a graph of order n whose non-zero eigenvalues are given by $\left\{{\lambda }_i\right\},1\le i\le r,$ for some $r\le n,$ then the non-zero eigenvalues of the graph $H_G^{(2k,p)}$ derived from G and $M^{(2k,p)}$ are $\left\{-{\lambda }_i\right\},1\le i\le r,$ with multiplicity $(2k-2)p,$ $\left\{(k-1){\lambda }_i\right\},1\le i\le r,$ with multiplicity $2p-2,$ $\left\{(k-1\pm kp){\lambda }_i\right\}$ for $1\le i\le r,$ which are $2r$ number. Also, zero is an eigenvalue $H_G^{(2k,p)}$ with multiplicity $2kp(n-r).$

Proof. 

Since the characteristic polynomial of the adjacency matrix of multipartite graph $M^{(2k,p)}$ is given by

$$\begin{array}{cc}\hfill f_{M^{(2k,p)}}\left(\lambda \right)& =Det\left(\left[\begin{array}{ccccccc}\lambda I_p& -J_p& -I_p& -J_p& -I_p& \cdots & -J_p\\ -J_p& \lambda I_p& -J_p& -I_p& -J_p& \cdots & -I_p\\ -I_p& -J_p& \lambda I_p& -J_p& -I_p& \cdots & -J_p\\ -J_p& -I_p& -J_p& \lambda I_p& -J_p& \cdots & -I_p\\ -I_p& -J_p& -I_p& -J_p& \lambda I_p& \cdots & -J_p\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ -J_p& -I_p& -J_p& -I_p& -J_p& \cdots & \lambda I_p\end{array}\right]\right).\hfill \end{array}$$

Now, we will replace $J_p$ and $I_p$ with J and $I,$ respectively. According to Lemma 2,

$$\begin{array}{cc}\hfill & f_{M^{(2k,p)}}\left(\lambda \right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{ccccccc}\lambda I& -J& -I& -J& -I& \cdots & -I\\ -J& \lambda I& -J& -I& -J& \cdots & -J\\ -I& -J& \lambda I& -J& -I& \cdots & -I\\ -J& -I& -J& \lambda I& -J& \cdots & -J\\ -I& -J& -I& -J& \lambda I& \cdots & -I\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ -I& -J& -I& -J& \lambda I& \cdots & -I\end{array}\right]\right.\hfill \\ \hfill & \left.-{\displaystyle \frac{1}{\lambda }}\left[\begin{array}{c}-J\\ -I\\ -J\\ -I\\ ⋮\\ -J\end{array}\right]\left[\begin{array}{cccccc}-J& -I& -J& -I& \cdots & -J\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{ccccc}\lambda I-{\displaystyle \frac{p}{\lambda }}J& -(1+{\displaystyle \frac{1}{\lambda }})J& -I-{\displaystyle \frac{p}{\lambda }}J& \cdots & -I-{\displaystyle \frac{p}{\lambda }}J\\ -(1+{\displaystyle \frac{1}{\lambda }})J& (\lambda -{\displaystyle \frac{1}{\lambda }})I& -(1+{\displaystyle \frac{1}{\lambda }})J& \cdots & -(1+{\displaystyle \frac{1}{\lambda }})J\\ -I-{\displaystyle \frac{p}{\lambda }}J& -(1+{\displaystyle \frac{1}{\lambda }})J& \lambda I-{\displaystyle \frac{p}{\lambda }}J& \cdots & -I-{\displaystyle \frac{p}{\lambda }}J\\ ⋮& ⋮& ⋮& ⋮& ⋮\\ -I-{\displaystyle \frac{p}{\lambda }}J& -(1+{\displaystyle \frac{1}{\lambda }})J& -I-{\displaystyle \frac{p}{\lambda }}J& \cdots & \lambda I-{\displaystyle \frac{p}{\lambda }}J\end{array}\right]\right)\hfill \end{array}$$

Add the negative of the first row to each odd row and the negative of the second row to each even row.

$$\begin{array}{cc}\hfill & f_{M^{(2k,p)}}\left(\lambda \right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{ccccc}\lambda I-{\displaystyle \frac{p}{\lambda }}J& -(1+{\displaystyle \frac{1}{\lambda }})J& -I-{\displaystyle \frac{p}{\lambda }}J& \cdots & -I-{\displaystyle \frac{p}{\lambda }}J\\ -(1+{\displaystyle \frac{1}{\lambda }})J& (\lambda -{\displaystyle \frac{1}{\lambda }})I& -(1+{\displaystyle \frac{1}{\lambda }})J& \cdots & -(1+{\displaystyle \frac{1}{\lambda }})J\\ (-1-\lambda )I& 0& (\lambda +1)I& \cdots & 0\\ 0& (-1-\lambda )I& 0& \cdots & 0\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ 0& (-1-\lambda )I& 0& \cdots & 0\\ (-1-\lambda )I& 0& 0& \cdots & (\lambda +1)I\end{array}\right]\right)\hfill \end{array}$$

Add all the odd columns to the first column and all the even columns to the second column, then apply the Lemma 2 to obtain the following result.

$$\begin{array}{cc}\hfill & f_{M^{(2k,p)}}\left(\lambda \right)\hfill \\ \hfill & ={\lambda }^p{(\lambda +1)}^{(2k-3)p}Det\left(\left[\begin{array}{cc}(\lambda -(k-1))I-{\displaystyle \frac{kp}{\lambda }}J& -(k-1)(1+{\displaystyle \frac{1}{\lambda }})J\\ -k(1+{\displaystyle \frac{1}{\lambda }})J& (\lambda -{\displaystyle \frac{k-1}{\lambda }}-(k-2))I\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^p{(\lambda +1)}^{(2k-3)p}{\left(\lambda -{\displaystyle \frac{k-1}{\lambda }}-(k-2)\right)}^{p}Det\left((\lambda -(k-1))I-{\displaystyle \frac{kp}{\lambda }}J\right.\hfill \\ \hfill & \left.-{\left(\lambda -{\displaystyle \frac{k-1}{\lambda }}-(k-2)\right)}^{-1}k(k-1)p{(1+{\displaystyle \frac{1}{\lambda }})}^{2}J\right)\hfill \\ \hfill & ={\lambda }^p{(\lambda +1)}^{(2k-3)p}{\left({\displaystyle \frac{2{\lambda }^2-(2k-4)\lambda -(2k-2)}{2\lambda }}\right)}^p{\left(\lambda -(k-1)\right)}^{p-1}\hfill \\ \hfill & \left[\lambda -(k-1)-{\displaystyle \frac{k{p}^2}{\lambda }}-{\displaystyle \frac{4k(k-1)p^2{(\lambda +1)}^2}{2\lambda \left(2{\lambda }^2-2(k-2)\lambda -2(k-1)\right)}}\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\lambda }^p{(\lambda +1)}^{(2k-3)p}{\left({\displaystyle \frac{{\lambda }^2-(k-2)\lambda -(k-1)}{\lambda }}\right)}^p{\left(\lambda -k+1\right)}^{p-1}\hfill \\ \hfill & \left[\lambda -k+1-{\displaystyle \frac{k{p}^2}{\lambda }}-{\displaystyle \frac{2k(2k-2)p^2{(\lambda +1)}^2}{2\lambda \left(2{\lambda }^2-2(k-2)\lambda -2(k-1)\right)}}\right]\hfill \\ \hfill & ={\left(\lambda +1\right)}^{(2k-3)p}{\left[{\lambda }^2-(k-2)\lambda -(k-1)\right]}^{p-1}{\left[\lambda -(k-1)\right]}^{p-1}\left(\lambda +1\right)\hfill \\ \hfill & \left(\lambda -k-kp+1\right)\left(\lambda -k+kp+1\right)\hfill \\ \hfill & ={\left(\lambda +1\right)}^{(2k-2)p}{\left[\lambda -(k-1)\right]}^{2p-2}\left(\lambda -k-kp+1\right)\left(\lambda -k+kp+1\right)\hfill \end{array}$$

So, the non-zero eigenvalues of the matrix $A_{M^{(2k,p)}}$ are $-1$ with multiplicity $(2k-2)p,$ $k-1$ with multiplicity $2p-2,k-1\pm kp.$ Let $A_G$ and $A_{H_G^{(2k,p)}}$ are the adjacency matrices of the graphs G and $H_G^{(2k,p)}$ respectively, then $A_{H_G^{(2k,p)}}=A_{M^{(2k,p)}}\otimes A_G$. Then from Lemma 3 the theorem follows. □

Similarly, we have the following corollary when $k=2.$

Corollary 2. 

If G is a graph of order n whose non-zero eigenvalues are given by $\left\{{\lambda }_i\right\},1\le i\le r,$ for some $r\le n,$ then the non-zero eigenvalues of the graph $H_G^{(4,p)}$ derived from G and $M^{(4,p)}$ are $\left\{-{\lambda }_i\right\},1\le i\le r,$ with multiplicity $2p,$ $\left\{{\lambda }_i\right\},1\le i\le r,$ with multiplicity $2p-2,$ $\left\{(\lambda -1\pm 2p){\lambda }_i\right\}$ for $1\le i\le r,$ which are $2r$ number. Also, zero is an eigenvalue $H_G^{(4,p)}$ with multiplicity $4p(n-r).$

Proof. 

The characteristic polynomial of the adjacency matrix of multipartite graph $M^{(4,p)}$ is given by

$$\begin{array}{cc}\hfill f_{M^{(4,p)}}\left(\lambda \right)& =Det\left(\left[\begin{array}{cccc}\lambda I_{p\times p}& -J_{p\times p}& -I_{p\times p}& -J_{p\times p}\\ -J_{p\times p}& \lambda I_{p\times p}& -J_{p\times p}& -I_{p\times p}\\ -I_{p\times p}& -J_{p\times p}& \lambda I_{p\times p}& -J_{p\times p}\\ -J_{p\times p}& -I_{p\times p}& -J_{p\times p}& \lambda I_{p\times p}\end{array}\right]\right).\hfill \end{array}$$

Nextly, we will substitute $J_{p\times p}$ with J and $I_{p\times p}$ with $I,$ respectively. According to Lemma 2,

$$\begin{array}{cc}\hfill & Det\left(\left[\begin{array}{cccc}\lambda I& -J& -I& -J\\ -J& \lambda I& -J& -I\\ -I& -J& \lambda I& -J\\ -J& -I& -J& \lambda I\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{ccc}\lambda I& -J& -I\\ -J& \lambda I& -J\\ -I& -J& \lambda I\end{array}\right]-\left[\begin{array}{c}J\\ I\\ J\end{array}\right]{\displaystyle \frac{I_p}{\lambda }}\left[\begin{array}{ccc}J& I& J\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{ccc}\lambda I& -J& -I\\ -J& \lambda I& -J\\ -I& -J& \lambda I\end{array}\right]-{\displaystyle \frac{1}{\lambda }}\left[\begin{array}{c}J\\ I\\ J\end{array}\right]\left[\begin{array}{ccc}J& I& J\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{ccc}\lambda I& -J& -I\\ -J& \lambda I& -J\\ -I& -J& \lambda I\end{array}\right]-{\displaystyle \frac{1}{\lambda }}\left[\begin{array}{ccc}pJ& J& pJ\\ J& I& J\\ pJ& J& pJ\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{ccc}\lambda I-{\displaystyle \frac{p}{\lambda }}J& (-1-{\displaystyle \frac{1}{\lambda }})J& -I-{\displaystyle \frac{p}{\lambda }}J\\ (-1-{\displaystyle \frac{1}{\lambda }})J& (\lambda -{\displaystyle \frac{1}{\lambda }})I& (-1-{\displaystyle \frac{1}{\lambda }})J\\ -I-{\displaystyle \frac{p}{\lambda }}J& (-1-{\displaystyle \frac{1}{\lambda }})J& \lambda I-{\displaystyle \frac{p}{\lambda }}J\end{array}\right]\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{ccc}\lambda I-{\displaystyle \frac{p}{\lambda }}J& (-1-{\displaystyle \frac{1}{\lambda }})J& -I-{\displaystyle \frac{p}{\lambda }}J\\ (-1-{\displaystyle \frac{1}{\lambda }})J& (\lambda -{\displaystyle \frac{1}{\lambda }})I& (-1-{\displaystyle \frac{1}{\lambda }})J\\ (-1-\lambda )I& 0& (\lambda +1)I\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^{p}Det\left(\left[\begin{array}{ccc}(\lambda -1)I-{\displaystyle \frac{2p}{\lambda }}J& (-1-{\displaystyle \frac{1}{\lambda }})J& -I-{\displaystyle \frac{p}{\lambda }}J\\ -2(1+{\displaystyle \frac{1}{\lambda }})J& (\lambda -{\displaystyle \frac{1}{\lambda }})I& (-1-{\displaystyle \frac{1}{\lambda }})J\\ 0& 0& (\lambda +1)I\end{array}\right]\right)\hfill \\ \hfill & ={\lambda }^p{(\lambda +1)}^{p}Det\left(\left[\begin{array}{cc}(\lambda -1)I-{\displaystyle \frac{2p}{\lambda }}J& (-1-{\displaystyle \frac{1}{\lambda }})J\\ -2(1+{\displaystyle \frac{1}{\lambda }})J& (\lambda -{\displaystyle \frac{1}{\lambda }})I\end{array}\right]\right)\hfill \\ \hfill & ={(\lambda +1)}^p{({\lambda }^2-1)}^{p}Det\left((\lambda -1)I-{\displaystyle \frac{2p}{\lambda }}J-2{\left({\displaystyle \frac{\lambda }{{\lambda }^2-1}}\right)}^2{\displaystyle \frac{p{(\lambda +1)}^2}{{\lambda }^2}}J\right)\hfill \\ \hfill & ={(\lambda +1)}^p{\left({\lambda }^2-1\right)}^{p}Det\left((\lambda -1)I-{\displaystyle \frac{4p}{\lambda -1}}J\right)\hfill \\ \hfill & ={(\lambda +1)}^p{\left({\lambda }^2-1\right)}^p{(\lambda -1)}^{p-1}\left[\lambda -1-{\displaystyle \frac{4p^2}{\lambda -1}}\right]\hfill \\ \hfill & ={(\lambda +1)}^{2p}{\left(\lambda -1\right)}^{2p-2}(\lambda -1+2p)(\lambda -1-2p).\hfill \end{array}$$

So, the non-zero eigenvalues of the matrix $A_{M^{(4,p)}}$ are $-1$ with multiplicity $2p,$ 1 with multiplicity $2p-2,1\pm 2p.$ Let $A_G$ and $A_{H_G^{(4,p)}}$ are the adjacency matrices of the graphs G and $H_G^{(4,p)}$ respectively, then $A_{H_G^{(4,p)}}=A_{M^{(4,p)}}\otimes A_G$. Then from Lemma 3 the corollary follows. □

5. Applications of Multipartite Graph Operations

In this section, we will discuss several applications of the multipartite graph operations in the construction of integral, orderenergetic and equienergetic graphs.

5.1. Applications of Graph Operation Based on Multipartite Graph with Odd Order

From this kind of multipartition graphs with odd order, we can characterize some of the derived graphs which are integral graphs.

Theorem 3. 

Let G be an integral graph of order $n.$ Then the graph $H_G^{(2k+1,p,q)}$ derived from $M^{(2k+1,p,q)}$ and G is an integral graph if $pq(k^2+k)=s^2+s,$ for some positive integer $s.$

Proof. 

Let G be an integral graph of order $n.$ Then from Theorem 1, $H_G^{(2k+1,p,q)}$ is also an integral graph if ${\displaystyle \frac{(2k-1)\pm \sqrt{4k^{2}pq+4kpq+1}}{2}}$ are integers. This will happen only when $4k^{2}pq+4kpq+1$ is a perfect square. Let $pq(k^2+k)=f\left(s\right),$ where $f\left(s\right)$ is non-negative integer valued function. Then $1+4pq(k^2+k)=1+4f\left(s\right).$

The general integer value function making this a perfect square is $f\left(s\right)=s(s+1),$ so that $1+4pq(k^2+k)=1+4s(s+1)={(2s+1)}^2.$ So the required value of $pq(k^2+k)=s(s+1).$ Then the eigenvalues of $M^{(2k+1,p,q)}$ become $k+s$ and $k-s-1,$ for some positive integer $s.$ So, from Theorem 1 it follows that the eigenvalues of $H_G^{(2k+1,p,q)}$ are integers and so $H_G^{(2k+1,p,q)}$ is an integral graph if G is an integral graph. □

The next theorem gives a general method for generating orderenergetic graphs from a given graph.

Theorem 4. 

Let G be any graph on n vertices with integral energy $\mathcal{E}\left(G\right).$ If $s,k$ are all positive integers satisfying

$$\mathcal{E}\left(G\right)={\displaystyle \frac{\left(p\right(k+1)+qk)n}{(2p-1)k+(2q-1)(k-1)+2s+1}},$$

(2)

then the graph $H_G^{(2k+1,p,q)}$ derived from $M^{(2k+1,p,q)}$ and G with $pq(k^2+k)=s^2+s$ is an orderenergetic graph.

Proof. 

If the $s,k$ are all positive integers satisfying Equation (2) where $\mathcal{E}\left(G\right)$ is the integral energy of the graph $G.$ Let $A_{H_G^{(2k+1,p,q)}}$ be the adjacency matrix of the graph $H_G^{(2k+1,p,q)}$ with $pq(k^2+k)=s^2+s.$ Then it follows that

$$\mathcal{E}\left(H_G^{(2k+1,p,q)}\right)=\left[(2p-1)k+(2q-1)(k-1)+2s+1\right]\mathcal{E}\left(G\right),$$

using Theorem 1. Now, using Equation (2), we have

$$\begin{array}{cc}\hfill \mathcal{E}\left(H_G^{(2k+1,p,q)}\right)& =\left[(2p-1)k+(2q-1)(k-1)+2s+1\right]\mathcal{E}\left(G\right)\hfill \\ \hfill & =\left(p\right(k+1)+qk)n,\hfill \end{array}$$

where $\left(p\right(k+1)+qk)n$ is the order of the graph $H_G^{(2k+1,p,q)}.$ So the graph $H_G^{(2k+1,p,q)}$ is an orderenergetic graph. □

Similarly, we have the following corollary when $k=1.$

Corollary 3. 

Let G be an integral graph of order $n.$ Then the graph $H_G^{(3,p,q)}$ derived from $M^{(3,p,q)}$ and G is an integral graph when $pq={\displaystyle \frac{1}{2}}s(s+1),$ for any positive integer $s.$

Corollary 4. 

Let G be any graph on n vertices with integral energy $\mathcal{E}\left(G\right).$ If s is a positive integer satisfying

$$\mathcal{E}\left(G\right)={\displaystyle \frac{(s^2+s+4p^2)n}{4p(p+s)}}.$$

(3)

Then the graph $H_G^{(3,p,q)}$ derived from $M^{(3,p,q)}$ and G with $pq={\displaystyle \frac{1}{2}}s(s+1)$ is always an orderenergetic graph.

Based on the aforementioned theorem and corollary, we can generate numerous orderenergetic graphs and equienergetic graphs. The following are some of the results.

Corollary 5. 

If G is an orderenergetic graph on n vertices, then the sequence of graphs $H_G^{(3,p,8p-2)}$ derived from $M^{(3,p,8p-2)}$ and G are also orderenergetic graphs with order $(10p-2)n$ for any positive integer $p.$

Proof. 

The result follows from Corollary 4 by setting $s=4p-1$ and noting that

$$\mathcal{E}\left(G\right)={\displaystyle \frac{(s^2+s+4p^2)}{4p(p+s)}}n={\displaystyle \frac{(4p(4p-1)+4p^2)}{4p(p+4p-1)}}n={\displaystyle \frac{5p-1}{5p-1}}n=n.$$

□

Corollary 6. 

If G is an orderenergetic graph on n vertices, then the sequence of graphs $H_{Sp{l}_{20}\left(G\right)}^{(7,24,14)}$ derived from $Sp{l}_{20}\left(G\right)$ and $M^{(7,24,14)}$ are orderenergetic graphs with order $2898n$.

Proof. 

From Theorem 1, we have

$$\mathcal{E}\left(H_{Sp{l}_{20}\left(G\right)}^{(7,24,14)}\right)=\mathcal{E}\left(M^{(7,24,14)}\right)\mathcal{E}\left(Sp{l}_{20}\left(G\right)\right)=322\times \left(9n\right)=2898n,$$

where $2898n$ is the order of the graph $H_{Sp{l}_{20}\left(G\right)}^{(7,24,14)}.$ So the graph $H_{Sp{l}_{20}\left(G\right)}^{(7,24,14)}$ is an orderenergetic graph. □

Similarly, the following corollary can be obtained.

Corollary 7. 

The sequence of graphs $H_{Sp{l}_6\left(K_{n_1,4n_1}\right)}^{(5,2,13)}$ derived from $Sp{l}_6\left(K_{n_1,4n_1}\right)$ and $M^{(5,2,13)}$ are orderenergetic graphs with order $1120n_1$ for any positive integer $n_1.$

These multipartite graph operations can also be used to generate equienergetic graphs.

Corollary 8. 

Let G be any graph, then the pair of graphs $H_G^{(5,p,2)}$ and $H_G^{(7,1,p)}$ are non-cospectral equienergetic graphs of same order, for any positive integer $p.$ Moreover if the graph G is an integral graph and $p={\displaystyle \frac{1}{12}}s(s+1)$ is a positive integer for some positive integer $s,$ then all the pairs of equienergetic graphs are also integral graphs.

Corollary 9. 

Let G and $G^{\prime }$ be a pair of equienergetic graphs, then the graphs $H_G^{(2k+1,p,q)}$ and $H_{G^{\prime }}^{(2k+1,p,q)}$ are also pair of equienergetic graphs for any positive integer $k,p,q.$

5.2. Applications of Graph Operation Based on Multipartite Graph with Even Order

According to the Theorem 2, we know the multipartite graphs $M^{(2k,p)}$ are all integral graphs for any positive integer $k,p.$ From this kind of multipartition graphs with even order, we can characterize some of the graphs which are orderenergetic graphs.

Theorem 5. 

Let G be any graph on n vertices with integral energy $\mathcal{E}\left(G\right).$ If k is a positive integer satisfying

$$\mathcal{E}\left(G\right)={\displaystyle \frac{kpn}{3kp-2p-k+1}},$$

(4)

then the graph $H_G^{(2k,p)}$ derived from $M^{(2k,p)}$ and G is an orderenergetic graph.

Proof. 

If the k is a positive integer satisfying Equation (4), where $\mathcal{E}\left(G\right)$ is the integral energy of the graph $G.$ Let $A_{H_G^{(2k,p)}}$ be the adjacency matrix of the graph $H_G^{(2k,p)}.$ Then it follows that

$$\mathcal{E}\left(H_G^{(2k,p)}\right)=\left(6kp-4p-2k+2\right)\mathcal{E}\left(G\right).$$

Now, using Equation (4), we have

$$\begin{array}{c}\hfill \mathcal{E}\left(H_G^{(2k,p)}\right)=\left(6kp-4p-2k+2\right)\mathcal{E}\left(G\right)=2kpn,\end{array}$$

where $2kpn$ is the order of the graph $H_G^{(2k,p)}.$ So the graph $H_G^{(2k,p)}$ is an orderenergetic graph. □

Similarly, we also have the following corollary when $k=2.$

Corollary 10. 

Let G be any graph on n vertices with integral energy $\mathcal{E}\left(G\right).$ If $\mathcal{E}\left(G\right)={\displaystyle \frac{2pn}{4p-1}},$ then the graph $H_G^{(4,p)}$ derived from $M^{(4,p)}$ and G is an orderenergetic graph.

Based on the aforementioned theorem and corollary, we can generate numerous orderenergetic graphs and equienergetic graphs. The following are some of the results.

Corollary 11. 

The sequence of graphs $H_{Sp{l}_6\left(K_{n_1,4n_1}\right)}^{(4,2)}$ derived from $Sp{l}_6\left(K_{n_1,4n_1}\right)$ and $M^{(4,2)}$ are orderenergetic graphs with order $280n_1$ for any positive integer $n_1.$

Proof. 

The result follows from Corollary 10

$$\mathcal{E}\left(H_{Sp{l}_6\left(K_{n_1,4n_1}\right)}^{(4,2)}\right)=\mathcal{E}\left(M^{(4,2)}\right)\mathcal{E}\left(Sp{l}_6\left(K_{n_1,4n_1}\right)\right)=14\times 20n_1=280n_1,$$

where $280n_1$ is the order of the graph $H_{Sp{l}_6\left(K_{n_1,4n_1}\right)}^{(4,2)}.$ So the graph $H_{Sp{l}_6\left(K_{n_1,4n_1}\right)}^{(4,2)}$ is an orderenergetic graph. □

Similarly, the following results can be obtained.

Corollary 12. 

If G is an orderenergetic graph on n vertices, then the sequence of graphs $H_{Sp{l}_{30}\left(G\right)}^{(26,33)}$ derived from $Sp{l}_{30}\left(G\right)$ and $M^{(26,33)}$ are orderenergetic graphs with order $26598n$.

Corollary 13. 

The sequence of graphs $H_{Sp{l}_6\left(K_{n_1,9n_1}\right)}^{(10,3)}$ derived from $Sp{l}_6\left(K_{n_1,9n_1}\right)$ and $M^{(10,3)}$ are orderenergetic graphs with order $2100n_1$ for any positive integer $n_1.$

These multipartite graph operations can also be used to generate equienergetic graphs. The following results can be obtained by using Theorem 2 to calculate the energy of the derived graph.

Corollary 14. 

Let G and $G^{\prime }$ be a pair of equienergetic graphs, then the graphs $H_G^{(2k,p)}$ and $H_{G^{\prime }}^{(2k,p)}$ are also pair of equienergetic graphs for any positive integer $k,p.$ Moreover, if the graphs G and $G^{\prime }$ are integral graphs, then all the pair of equienergetic graphs are also integral graphs.

Corollary 15. 

Let G and $G^{\prime }$ be two orderenergetic graphs with order $6n_1$ and $5n_1,$ respectively. Then the graphs $H_G^{(10,3)}$ and $H_{G^{\prime }}^{(18,2)}$ are a pair of equienergetic graphs.

Corollary 16. 

If $p={\displaystyle \frac{1}{8}}(3k+1)$ is a positive integer, then the graphs $H_{Sp{l}_2\left(K_{n_1,5n_1}\right)}^{(2k,2p)}$ and $H_{K_{2n_1,10n_1}}^{(6k,p)}$ are a pair of equienergetic and non-cospectral graphs for some positive integer k and any positive integer $n_1.$

Remark 1. 

Similar to the proofs of Theorems 4 and 5, other special graphs related to eigenvalues or energy can be obtained, such as borderenergetic graph, hyperenergetic graph, hypoenergetic graph.

Remark 2. 

It is worth noting that we can conclude that the energy of the graph $M^{(4,p)}$ is $2p+2p-2+1+2p+2p-1=8p-2$, while the number of vertices in the graph $M^{(4,2)}$ is $4p$, which demonstrates that the graph $M^{(4,p)}$ is exactly a borderenergetic graph.

6. Conclusions

In recent years, Joseph, S. P. proposed several graph operations to generate the special graphs in [25,26]. This paper primarily presents a new class of non-complete multipartite graph operations. These operations can be used to generate special graphs, such as infinitely many integral graphs, orderenergetic graphs, and infinitely many pairs of equienergetic graphs.

In the present work, the special graph related to eigenvalues or energy was mainly constructed by regular graph, complete graphs and complete multipartite graphs as well as their complement. In this paper, the non-complete multipartition graph was used to construct these special graphs. Even though such multipartite graphs are regular when m is even, the graph operations used in our construction are different, leading to inconsistent results.

In future work, we will extend our research in the following areas:

(i)

In [28], the authors introduce somen new concepts and operations of type 2 soft graphs. These concepts and operations may assist us in further exploring the construction and properties of these special graphs.

(ii)

Furthermore, they characterize type 2 soft graphs on underlying subgraphs of a simple graph in [28]. Investigating the type 2 soft graphs on underlying subgraphs of multipartite graphs is also one of our future research directions.

(iii)

Only one new class of non-complete multipartite graph and its associated three parameters are considered in this paper. Further research will explore additional incomplete multipartite graphs and related parameters.