1. Introduction
Spectral graph theory is concerned with the study of the eigenvalues associated with various matrices for a graph and how the eigenvalues relate to the structural characteristics of the graph. The spectrum of a graph is related to properties such as connectedness, diameter, independence number, chromatic number, and regularity. Many scholars have investigated the spectral characteristics of (adjacency matrices of) graphs; see [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11]. Let
G be the graph, such that
and
be the adjacency matrix of graph
G, which is square and symmetric; hence, its eigenvalues are all real. A singular graph
G is a graph for which
, otherwise, it is referred to as non-singular. The characteristic polynomial
of graph
G can be obtained from
. The eigenvalues of
, or simply the eigenvalues of graph
G, are the roots of this polynomial. The spectrum (or the multiset of all eigenvalues) of graph
G throughout the text is defined as
where
, and
is the multiplicity of each
for
. It is well known that graph
G is bipartite if and only if its nonzero eigenvalues are symmetric around 0. The sum of all absolute eigenvalues of graph
G is called the energy of a graph, denoted by
.
Definition 1 ([12]). The anti-reciprocal eigenvalue property (or property ) is said to hold by graph G if for each there exists . The multiplicities of the eigenvalues and their negative reciprocals may or may not be equal for graph G with property ; however, if each eigenvalue and its negative reciprocal have the same multiplicities, graph G is said to satisfy a strong anti-reciprocal eigenvalue (or property ).
Definition 2 ([13]). The reciprocal eigenvalue property (or property ) is said to hold by graph G if for each there exists . The multiplicities of the eigenvalues and their reciprocals may or may not be equal for graph G with property ; however, if each eigenvalue and its reciprocal have the same multiplicities, then that graph G is said to satisfy the strong reciprocal eigenvalue (or property ).
Note that if graph G is a bipartite graph, then properties and are symmetric around 0. The adjacency matrix of a simple connected graph is symmetric. This fact is useful for converting the adjacency matrix into a block matrix. Furthermore, this block form of the adjacency matrix is useful for computing the characteristic polynomials of the graph.
Graph G satisfying property (respectively, property ) is referred to as the reciprocal graph (respectively, strong reciprocal graph), and abbreviated as the graph (respectively, graph). Graph G satisfying property (respectively, property ) is referred to as an anti-reciprocal graph (respectively, strong anti-reciprocal graph), and abbreviated as graph (respectively, graph).
It is well known that graph
G is bipartite if and only if, for each eigenvalue
, there exits
in the spectrum of
G. In 1978, the authors investigated property
for non-singular trees in [
14] and [
15], respectively, but with different names, i.e., “symmetric property” and “property
C”, respectively. Later, Barik et al. [
16] renamed this property as property
in 2006 and introduced property
. They proved that properties
and
are equal in the case of nonsingular trees. Researchers investigated these properties for weighted trees in [
17] and a subclass of connected bipartite graphs (with a unique perfect matching) in [
18]. They showed that if we apply appropriate limitations on weight functions, these two properties are equal; however, in general, these properties are not the same, see [
19]. Unicyclic graphs with property
were studied in [
20]. It is worth noting that the study of reciprocal eigenvalue properties is strongly related to the concepts of ‘matching’ and ‘corona product’, which are both widely studied disciplines.
Definition 3 ([21]). Consider two simple connected graphs, i.e., and of orders and , respectively. The corona product of graphs and is a graph constructed with the help of one copy of graph and -copies of and then connecting each vertex of the copy of with the vertex of , where .
In 2012, Lagrange [
12] introduced the strong anti-reciprocal eigenvalue property for graphs and investigated this property for zero-divisor graphs of finite commutative rings with non-zero divisors. In [
22], the authors investigated a family of graphs with a unique perfect matching
M, where the diagonal entries of the inverse of the adjacency matrix of each graph were all zero. Moreover, it was proved that each non-corona graph in this class did not satisfy the property
, even for a single weight function
.
In 2017, Ahmad et al. investigated a class of weighted graphs(
) with a unique perfect matching
M [
23]. They proved that the weighted graph
of a graph
satisfies property
for all
if and only if
G is a corona graph.
In [
24], the authors raised the question of “whether non-corona graphs with the property
exist?” and then answered this question by constructing seven types of unweighted non-corona graphs that satisfy property
. These constructions were later generalized by Barik et al. in [
25] for unweighted graphs; in [
26], the authors investigated property
for the generalized families constructed in [
25] with a specific weight function. In [
27], the authors investigated some new families of non-corona graphs with property
.
The study of graph theory has led to significant advancements in various fields, ranging from computer science to social networks. However, there is a constant pursuit to uncover new families of graphs that possess unique properties and offer fresh insights into the underlying structures of complex networks. In this vein, a novel family of graphs, called flabellum graphs, was introduced. By investigating and analyzing the properties of flabellum graphs, such as their connectivity, energy, and degree sequences, the aim is to contribute to the ever-growing body of knowledge in graph theory. Furthermore, the construction of several families of graphs can be facilitated using the flabellum graph network. Additionally, by evaluating the energies associated with these graphs, we are exploring the intricate relationship between their structural attributes and their physical properties. The following concepts will be used in the next section.
Definition 4. The Dutch windmill graph , , , is the graph obtained by taking k copies of the cycle with a vertex in common.
Lemma 1 ([23]). A polynomial is said to satisfy property if and only if The following lemmas will be used to prove our main results.
Lemma 2 ([28]). Let A be an matrix and Then for any constant c, Lemma 3 ([2]). Let be a block matrix, where and are square matrices. Then Lemma 4 ([2]). Let A be a block matrix, i.e., where and are square matrices. Let be an invertible square matrix. Then A is invertible if and only if the Schur complement S of is invertible, i.e., is invertible, and Remark 1 ([27]). Let and be two polynomials of degrees and , respectively. If both polynomials satisfy property , then the polynomial satisfies property for any constant c.
2. Main Results
In this section, we introduce certain families of graphs known as flabellum graphs, flabellum cycle graphs, flabellum complete graphs, and flabellum star graphs. Then with the help of these new families of graphs, several families of graphs that are non-bipartite and non-corona are constructed, and with the help of several algebraic techniques, we prove that the graphs in these families are graphs. Moreover, the energy of the flabellum graph is calculated.
Definition 5. Consider the cycle graph with copies, where each copy has a common vertex known as the central vertex. In j copies of the cycle graph, we add an edge between the central vertex and each vertex that is not adjacent to it, where . This results in the flabellum graph , which can be seen in Figure 1. The order and size of the flabellum graph are and , respectively. For , the flabellum graph is isomorphic to either the Dutch windmill graph or the friendship graph. Theorem 1. Let be a graph, then Proof. Since the adjacency matrix of the graph
is symmetric, it can be written in the block form as follows:
Then
by Lemma 3, we obtain
where
,
and
Then
Remark 2. is very close to satisfying property .
Example 1. The graph shown in Figure 2 is very close to satisfying the property . Here, . Then We will now define some generalized flabellum graphs, namely the flabellum complete graph, flabellum cycle graph, and flabellum star graph.
Definition 6. Consider a complete graph and m copies of the flabellum graph . The flabellum complete graph can be obtained by attaching a copy of the flabellum graph to each vertex of the complete graph , as shown in Figure 3. Now, with the help of the flabellum complete graph we construct four different types of families of graphs, namely, , , , and .
Family
Consider the flabellum complete graph
. The graph
can be obtained by adding a pendant edge to each vertex of
. The family of all such graphs is denoted by
and
In the following theorem, it is established that each graph in the family is a graph.
Theorem 2. Let , then is a () graph.
Proof. Let
, then
. The adjacency matrix of
can be written as:
where
and
Then
by Lemma 3, we obtain
where
. Now, by using Lemma 2
and for
Here, and satisfy property . Moreover, satisfies property for . Thus, the characteristic polynomial satisfies property from Remark 1. Hence, graph is a graph. □
Example 2. Let , as shown in Figure 4. Then Therefore, graph is a graph.
Using the Laplace expansion, we have the following Lemma.
Lemma 5. Let A be an matrix, then for any constant c, Here, is the submatrix of A, obtained by deleting the first row and first column.
Family :
Consider , where k is an even integer, then the graph can be obtained by adding a pendant to each vertex of the complete graph of . The graph can be obtained by removing copies of the flabellum graph from vertices of . The family of all such graphs is denoted by . The following result shows that each graph in is a graph.
Theorem 3. Let . Then is a () graph.
Proof. Let
, then
. The adjacency matrix of
can be written as:
where
Then
by Lemma 3, we obtain
where
,
and
Hence, satisfies property . Consequently, graph is a graph. □
Example 3. Let , as shown in Figure 5. Then Therefore, graph is a graph.
The family of graphs can be generalized as follows.
Family :
Again, consider the flabellum complete graph (here, k is an even integer) and let be the graph obtained by adding a pendant to each vertex of of . The graph can then be obtained by removing copies of the flabellum graph from vertices of . The family of all such graphs is denoted by . The following result shows that each graph in is a property graph.
Theorem 4. Let , then is a () graph.
Proof. Let
, then
. The adjacency matrix of
can be written as:
where
and
Then
by Lemma 3, we obtain
where
. Now, by using Lemma 2
where
and for
Notice that and satisfy property . Moreover, satisfies property for . Therefore, according to Remark 1, the polynomial satisfies property . Hence, graph is a graph. □
Example 4. Let , as shown in Figure 6. Then Hence, graph is a graph.
The family of graphs can be generalized as follows.
Family :
Consider , which is the graph obtained by adding a pendant to each vertex of of . The graph can then be obtained by removing copies of the flabellum graph from vertices of , where is any positive integer. The family of these graphs is denoted by . We present the following theorem, which can be proven using similar steps as in the proofs of previous theorems.
Theorem 5. Let then is a () graph.
Definition 7. Consider a cycle graph and m copies of the flabellum graph . The flabellum cycle graph can be obtained by attaching a copy of the flabellum graph to each vertex of the cycle graph , as shown in Figure 7. Now, with the help of the flabellum cycle graph , we construct different families of strong anti-reciprocal graphs, namely, , , , and , which are defined as follows:
Family :
Consider the flabellum cycle graph
. The graph
can be obtained by adding a pendant edge to each vertex of
and the family of all such graphs is denoted by
and
Family :
Consider
, where
k is an even integer, then the graph
can be obtained by adding a pendant to each vertex of the cycle graph
of
. The graph
can be obtained by removing
copies of the flabellum graph from vertices of
. The family of all such graphs is denoted by
and
Family :
Again, consider the flabellum cycle graph
(here,
k is an even integer), and let
be the graph obtained by adding a pendant to each vertex of
of
. The graph
can then be obtained by removing
copies of the flabellum graph from vertices of
. The family of all such graphs is denoted by
and
The proof of the following theorem is similar to the proofs of Theorems 2–4.
Theorem 6. Let then is a () graph.
The family can be generalized as follows.
Family :
Consider , which is the graph obtained by adding a pendant to each vertex of of . The graph can then be obtained by removing copies of the flabellum graph from vertices of , where is any positive integer. The family of all such graphs is denoted by . The following theorem shows that the family is a family of graphs.
Theorem 7. Let , then is a () graph.
Example 5. Let , as shown in Figure 8a. Then Therefore, graph is a graph.
Example 6. Let , as shown in Figure 8b. Then Therefore, graph is a graph.
Example 7. Let , as shown in Figure 9a. Then Hence, this graph is a graph.
Definition 8. Consider a star graph and m copies of the flabellum graph . The flabellum star graph can be obtained by attaching a copy of the flabellum graph to each vertex of the star graph , as shown in Figure 10. Now, with the help of the flabellum star graph , we construct different families of strong anti-reciprocal graphs in the following definitions.
Family :
Consider the flabellum star graph
. The graph
can be obtained by adding a pendant edge to each vertex of
and the family of all such graphs is denoted by
and
Family :
Consider
, where
k is an even integer, then the graph
can be obtained by adding a pendant to each vertex of the star graph
of
. The graph
can be obtained by removing
copies of the flabellum graph from the vertices of
. The family of all such graphs is denoted by
and
Family :
Again, consider the flabellum star graph
(here,
k is an even integer), and let
be the graph obtained by adding a pendant to each vertex of
of
. The graph
can then be obtained by removing
copies of the flabellum graph from vertices of
. The family of all such graphs is denoted by
and
The following theorem, which can be proved in a similar way to the proofs of Theorems 2–4, reveals that the families , , and are families of graphs.
Theorem 8. Let , then is a () graph.
Example 8. Let , as shown in Figure 11a. Then Therefore, graph is a graph.
Example 9. Let , as shown in Figure 11b. Then Therefore, graph is a graph.
Example 10. Let as shown in Figure 9b. Then Therefore, graph is a graph.
The family of graphs can be generalized as follows.
Family :
Consider , which is the graph obtained by adding a pendant edge to each vertex of in . The graph can then be obtained by removing copies of the flabellum graph from vertices of , where is any positive integer. The family of all such graphs is denoted by . We obtain the following theorem, which can be proved using similar steps as in the proofs of previous theorems.
Theorem 9. Let then is a () graph.
Remark 3. All families can be generalized if we consider any connected graph instead of , or , and all of these generalized families are families of graphs.