Structural and Spectral Properties of Chordal Ring, Multi-Ring, and Mixed Graphs

Abstract

The chordal ring (CR) graphs are a well-known family of graphs used to model some interconnection networks for computer systems in which all nodes are in a cycle. Generalizing the CR graphs, in this paper, we introduce the families of chordal multi-ring (CMR), chordal ring mixed (CRM), and chordal multi-ring mixed (CMRM) graphs. In the case of mixed graphs, we can have edges (without direction) and arcs (with direction). The chordal ring and chordal ring mixed graphs are bipartite and 3-regular. They consist of a number r (for $r\ge 1$) of (undirected or directed) cycles with some edges (the chords) joining them. In particular, for CMR, when $r=1$, that is, with only one undirected cycle, we obtain the known families of chordal ring graphs. Here, we used plane tessellations to represent our chordal multi-ring graphs. This allowed us to obtain their maximum number of vertices for every given diameter. Additionally, we computationally obtained their minimum diameter for any value of the number of vertices. Moreover, when seen as a lift graph (also called voltage graph) of a base graph on Abelian groups, we obtained closed formulas for the spectrum, that is, the eigenvalue multi-set of its adjacency matrix.

1. Introduction

Interconnection networks play a crucial role in the design of distributed computer systems, often represented by graphs where vertices symbolize nodes or processing elements, and edges denote communication links between them. See, for instance, Deo [1], and Bermond et al. [2]. The topology of these networks is a key factor influencing communication delay, throughput, and message routing efficiency. Understanding graph theory concepts such as adjacency, degree, diameter, and mean distance is essential in analyzing and optimizing these networks. Exploring topologies like rings, meshes, and hypercubes can offer efficient and reliable interconnection solutions for multi-computer systems. One of the simplest topologies for interconnection networks is the undirected or directed graphs in which each node is connected to two others, making up a bidirectional loop or cycle. The main drawbacks of cycles are their poor reliability (any link or processor failure disconnects the network) and low performance (some messages must travel along half or the whole ring to reach their destination). To overcome these problems, the cyclic topology is improved by the chordal ring graphs, which consist of a cycle with some additional links between nodes. Arden and Lee [3] proposed the chordal ring graphs for efficient and reliable multi-(micro)computer interconnection networks, which are real-world examples. If only one link is added to each node, the corresponding graphs are 3-regular. This is the case for the so-called ‘chordal ring networks’; see an example in Figure 1 (left). Chordal rings were first introduced by Coxeter [4]. Since then, the structure of these graphs has been extensively studied. For example, Arden and Lee [3] studied the problem of the maximization of the number of nodes for a given diameter, and Yebra et al. [5] found a relationship between certain types of plane tessellations, where the vertices are associated with regular polygons and chordal ring graphs (see also Morillo et al. [6] and Dalfó et al. [7]). This geometrical representation characterizes the graph and facilitates the study of some of its parameters, particularly those with distance-related parameters. In this paper, we show that chordal ring graphs are one of four closely related families of graphs constituted by (undirected or directed) cycles with chords. These are the chordal ring and multi-ring graphs (CR and CMR) and the chordal ring and multi-ring mixed graphs (CRM and CMRM).

With the chordal graphs, one finds a smaller diameter for any value of the number of vertices and a greater number of vertices for every given diameter, both with respect to the cycle graphs. Our research gap is to improve these results with the multi-ring chordal graphs and the ring and multi-ring mixed graphs. This unified approach allows us to represent all of them as quotient graphs of a ‘running bond’ infinite graph pattern. This name refers to a way of placing blocks used by bricklayers; see Reid [8] and Figure 2 and Figure 3. As shown on the left in the same figures, such quotient graphs are obtained when the infinite graph is taken modulo some integral matrix M, whose rows are the translation vectors defining the periodicity. More precisely, the vertices of the infinite graph are identified with integral vectors, and two vectors u and v represent the same vertex of the quotient graph if and only if $\mathit{u}\equiv \mathit{v}mod\mathit{M}$. That is, $\mathit{u}-\mathit{v}$ belongs to the lattice generated by the rows of M, and $\mathit{u}-\mathit{v}\in {\mathbb{Z}}^2\mathit{M}$; see Fiol [9]. This approach is valid for graphs with edges, that is, bidirectional links. It is also valid for mixed graphs with edges (bidirectional links) and arcs (one-directional links). Moreover, it can be applied to any value of the number of cycles (with at least one cycle) and any number of chords (with at least one chord). The chords are the additional edges that allow us to obtain a chordal ring graph from a cycle.

We show how to construct the chordal ring families from graphs admitting a voltage assignment, that is, with ‘weights’ on the arcs. In this respect, Gross introduced the following concepts in [10]. Let G be a group.

An (ordinary ) voltage assignment on the (di)graph $\Gamma =(V,E)$ is a mapping $\alpha :E\to G$ with the property that $\alpha \left(a^{-1}\right)={\left(\alpha \left(a\right)\right)}^{-1}$ for every arc $a\in E$. Thus, a voltage assigns an element $g\in G$ to each arc of the graph so that a pair of mutually reverse arcs a and $a^{-1}$, forming an undirected edge, receive mutually inverse elements g and $g^{-1}$. The graph $\Gamma$ and the voltage assignment $\alpha$ determine a new graph ${\Gamma }^{\alpha }$, called the lift of $\Gamma$, which is defined as follows. The vertex set of the lift is the Cartesian product $V^{\alpha }=V\times G$. Moreover, for every arc $a\in E$ from a vertex u to a vertex v for $u,v\in V$ (possibly, $u=v$) in $\Gamma$, and, for every $g\in G$, there is an arc $(a,g)\in E^{\alpha }$ from the vertex $(u,g)\in V^{\alpha }$ to the vertex $(v,g\alpha \left(a\right))\in V^{\alpha }$. Let us show the example illustrated in Figure 4. In this example, when the base graph with a voltage assignment on the arcs (drawn on the left) is applied to the group ${\mathbb{Z}}_5$, we obtain the Petersen graph (drawn on the right). Since the group is ${\mathbb{Z}}_5$, we have five copies (numbered from 0 to 4) of the vertices of the base graph. So, as the voltage of the pink edge is 2, we have to join the black vertices by adding $2mod5$ to each copy. We do the same for the other edges, each with its corresponding voltage.

One of this paper’s main contributions is finding the spectra of the different families of chordal ring graphs and mixed graphs. We recall the conjecture by Haemers [11] that states that almost all graphs are determined by their spectra. More precisely, among all non-isomorphic graphs on, at most, n vertices, the fraction of graphs that are not determined by their spectra approaches 1 when n approaches infinity. With this aim, we used a quotient-like matrix, introduced by Dalfó et al. [12], that fully represents a lifted digraph. The main advantage of this approach is that such a matrix has a size equal to the order of the base digraph. For a digraph $\Gamma =(V,E)$ with voltage assignment $\alpha$, if we deal with the case when the group G of the voltage assignments is cyclic (that is, $G={\mathbb{Z}}_k=\{0,1,\dots ,k-1\}$), then its polynomial matrix $\mathit{B}\left(z\right)$ is a square matrix indexed by the vertices of $\Gamma$. The matrix elements are complex polynomials in the quotient ring ${\mathbb{R}}_{k-1}\left[z\right]=\mathbb{R}\left[z\right]/\left(z^k\right)$, where $\left(z^k\right)$ is the ideal generated by the polynomial $z^k$. More precisely, each entry of $\mathit{B}\left(z\right)$ is fully represented by a polynomial of degree at most $k-1$, say ${\left(\mathit{B}\left(z\right)\right)}_{\mathit{u}\mathit{v}}=p_{\mathit{u}\mathit{v}}\left(z\right)={\alpha }_0+{\alpha }_{1}z+\cdots +{\alpha }_{k-1}{z}^{k-1}$, where

$${\alpha }_i=\left\{\begin{array}{cc}1\hfill & if uv\in E \mathrm{and} \alpha \left(uv\right)=i,\hfill \\ 0\hfill & \mathrm{otherwise}\hfill \end{array}\right.$$

for $i=0,\dots ,k-1$.

This paper is structured as follows. In the next section, we present our approach to chordal ring graphs and obtain their spectra when seeing them as lift graphs. In Section 3, we introduce the chordal multi-ring graphs to obtain graphs with an optimal number of vertices for a given diameter and find their spectra through voltage graphs. Similar results are obtained in Section 4 for the chordal ring mixed graphs (in which the cycle is directed and the chords are undirected) and, finally, in Section 5 for the chordal multi-ring mixed graphs. Our approach allows us to determine the spectra in all the cases.

2. Chordal Ring Graphs

The chordal ring graph $CR(N,c)$ has an even number of vertices $(N=2n)$ labeled with the integers $\{0,1,\dots ,N-1\}$, and each even vertex i is connected to the vertices $i\pm 1 \mathrm{mod} N$ and $i+c \mathrm{mod} N$ for some odd integer c. Consequently, each odd vertex j is connected to the vertices $j\pm 1 \mathrm{mod} N$ and $j-c \mathrm{mod} N$. See Figure 5 on the left. Therefore, we have a ring structure with additional links called chords. An example is the Heawood graph with a diameter 3, isomorphic to $CR(14,5)$, and it is known to be a $(3,6)$-cage; see Figure 1 (left). The chordal ring graphs are bipartite and vertex-symmetric, that is, they have an automorphism group that acts transitively on the vertices. We recall that a group of automorphisms is an algebraic structure that defines the symmetries in the graph. More precisely, an automorphism of a graph $G=(V,E)$ is a permutation $\sigma$ of the vertex set V, such that the pair of vertices $(u,v)$ forms an edge if and only if the pair $\left(\sigma \right(u),\sigma (v\left)\right)$ also forms an edge. In fact, this property is shared by the other three families studied in this paper: the multi-ring graphs $\left(CMR\right)$, and the chordal ring and multi-ring mixed graphs ($CRM$ and $CMRM$). More details about the symmetries of the chordal ring graphs are in the following result.

Proposition 1. 

$\left(i\right)$

The chordal ring graph $CR(N,c)$ is isomorphic to $CR(N,-c)$.

$\left(ii\right)$

The automorphism group of $CR(N,c)$, with $N=2n$, contains the dihedral group $D_n$ with $N=2n$ elements and presentation

$$D_n=\langle \sigma ,\tau |{\sigma }^n={\tau }^2={\left(\sigma \tau \right)}^2=e\rangle$$

(1)

where e denotes the identity element.

Proof. 

$\left(i\right)$

It should be noted that, in $CR(N,-c)$, each vertex i is adjacent to $i-c$ if i is even and $i+c$ if i is odd. Then, let us prove that the mapping $\gamma :i\mapsto i+1$ is an isomorphism from $CR(N,c)$ to $CR(N,-c)$:

–

In $CR(N,c)$, the vertex i is adjacent to $i\pm 1$, whereas, in $CR(N,-c)$, $\gamma \left(i\right)=i-1$ is adjacent to $\gamma (i\pm 1)=\{i,i-2\}$.

–

In $CR(N,c)$, the vertex i (even) is adjacent to $i+c$, whereas, in $CR(N,-c)$, $\gamma \left(i\right)=i-1$ (odd) is adjacent to $\gamma (i+c)=i-1+c$.

–

In $CR(N,c)$, the vertex i (odd) is adjacent to $i-c$, whereas, in $CR(N,-c)$, $\gamma \left(i\right)=i-1$ (even) is adjacent to $\gamma (i-c)=i-1-c$.

$\left(ii\right)$

Let us consider the mappings $\sigma :i\mapsto i+2$ and $\tau :i\mapsto -i-1$ from $CR(N,c)$ to itself. Then, let us first check the defining relations in (1): we have ${\sigma }^n\left(i\right)=i+2n=i$, ${\tau }^2\left(i\right)=\tau (-i-1)=-(-1-i)-1=i$, and

$${\left(\sigma \tau \right)}^2\left(i\right)=\sigma \tau \sigma (-i-1)=\sigma \tau (-i+1)=\sigma (i-2)=i.$$

In order to prove that $\sigma$ and $\tau$ are automorphisms, let us represent the vertices adjacent to i as $G_{\pm }\left(i\right)=\{i+1,i-1\}$ and $G_c\left(i\right)=i+c$ for i even, and $G_c\left(i\right)=i-c$ for i odd. Then, the following properties are obtained:

–

For every vertex i,

$$\sigma \left(G_{\pm }\left(i\right)\right)=\sigma (i\pm 1)=\{i+3,i+1\}=G_{\pm }(i+2)=G\left(\sigma \left(i\right)\right).$$

–

If i is even,

$$\sigma \left(G_c\left(i\right)\right)=\sigma (i+c)=i+2+c=G_c(i+2)=G_c\left(\sigma \left(i\right)\right).$$

–

If i is odd,

$$\sigma \left(G_c\left(i\right)\right)=\sigma (i-c)=i+2-c=G_c(i+2)=G_c\left(\sigma \left(i\right)\right).$$

Thus, in all the cases, $\sigma$ and $\tau$ commute with $G_{\pm }$ and $G_c$, proving that they are automorphisms of $CR(N,c)$. □

From the above result, if i and j are vertices with the same parity, then ${\sigma }^{{\displaystyle \frac{j-i}{2}}}\left(i\right)=j$. Otherwise, if i and j have different parity, ${\sigma }^{{\displaystyle \frac{i+j+1}{2}}}\tau \left(i\right)={\sigma }^{{\displaystyle \frac{i+j+1}{2}}}(-i-1)=j$. This shows that, as it was commented, there is always an automorphism mapping a vertex i to a vertex j.

As commented in the introduction, chordal ring graphs can be represented as congruent tiles that tessellate the plane periodically. More precisely, if each vertex of $CR(N,c)$ is represented by a numbered $modN$ unit square, the vertices reached at a distance $0,1,2,\dots$ from any given vertex can be arranged in a planar pattern, as shown in Figure 5, starting from vertex 0.

Since there are $3\ell$ vertices at a distance $\ell (>0)$ from vertex 0, and the graph is bipartite, it was shown (in Yebra et al. [5], and Morillo et al. [6]) that the maximum number $N\left(k\right)$ of vertices of a chordal ring with a diameter k is

$$N\left(k\right)=\left\{\begin{array}{cc}{\displaystyle \frac{3k^2+1}{2}}\hfill & \mathrm{for} k \mathrm{odd},\hfill \\ {\displaystyle \frac{3k^2}{2}}\hfill & \mathrm{for} k \mathrm{even}.\hfill \end{array}\right.$$

(2)

Moreover, it was shown that such a maximum can be attained when k is odd and cannot be attained when $k>2$ is even. Consequently, the following conjecture was raised in the same papers (see also Comellas and Hell [13]).

Conjecture 1. 

The maximum number N of vertices of a chordal ring graph with an even diameter $k>2$ is $N={\displaystyle \frac{3k^2}{2}}-k$.

The method used in [5,6] to obtain chordal rings with a maximum number of vertices $(3k^2+1)/2$ (for k odd) and $3k^2/2-k$ (for k even) consists of the following two steps:

(1)

Using the planar pattern, find the ‘optimal tiles’ (that is, containing the maximum number of vertices for a given diameter k), and check that they periodically tessellate the plane (see Figure 2 for $k=5$);

(2)

From the two basic translation vectors of the periodic tiling (generating the lattice of the positions of the vertices with label 0), solve a linear system of equations to find the chord c.

In

Table A1. The minimum diameter k for each number of vertices $N\le 528$ in the chordal ring graphs $CR(N,c)$ with (minimum) chord c. In boldface are the cases with a maximum number of vertices for a given diameter.

$\mathit{N}$	$\mathit{c}$	$\mathit{k}$	$\mathit{N}$	$\mathit{c}$	$\mathit{k}$	$\mathit{N}$	$\mathit{c}$	$\mathit{k}$	$\mathit{N}$	$\mathit{c}$	$\mathit{k}$	$\mathit{N}$	$\mathit{c}$	$\mathit{k}$	$\mathit{N}$	$\mathit{c}$	$\mathit{k}$
2	1	1	90	11	9	178	49	12	266	41	14	354	47	16	442	47	19
4	1	2	92	11	9	180	33	12	268	41	14	356	47	17	444	47	19
6	3	2	94	11	9	182	33	11	270	21	15	358	55	17	446	47	19
8	3	3	96	11	9	184	33	12	272	23	15	360	67	17	448	47	19
10	3	3	98	11	9	186	51	12	274	37	15	362	43	17	450	53	18
12	5	3	100	13	9	188	15	13	276	37	15	364	43	17	452	53	18
14	5	3	102	13	9	190	79	12	278	37	15	366	43	17	454	53	19
16	5	4	104	23	9	192	35	12	280	43	14	368	49	16	456	81	19
18	5	4	106	23	9	194	35	12	282	37	15	370	43	17	458	61	19
20	5	4	108	41	9	196	17	13	284	61	15	372	163	17	460	97	19
22	5	5	110	11	10	198	17	13	286	21	16	374	27	18	462	49	19
24	5	5	112	25	9	200	17	13	288	77	15	376	81	17	464	49	19
26	5	5	114	25	9	202	17	13	290	39	15	378	67	17	466	49	19
28	7	5	116	13	10	204	37	12	292	39	15	380	45	17	468	55	18
30	7	5	118	15	10	206	19	13	294	39	15	382	45	17	470	49	19
32	7	5	120	27	10	208	57	13	296	39	15	384	45	17	472	131	19
34	7	5	122	27	9	210	77	13	298	39	16	386	45	17	474	63	20
36	7	6	124	27	10	212	33	13	300	69	15	388	45	18	476	75	19
38	15	5	126	13	11	214	33	13	302	65	15	390	69	17	478	141	19
40	7	6	128	13	11	216	33	13	304	41	16	392	69	18	480	85	19
42	7	6	130	29	10	218	33	13	306	41	15	394	85	18	482	51	19
44	9	6	132	29	10	220	17	14	308	41	15	396	47	18	484	51	19
46	7	7	134	13	11	222	51	13	310	41	15	398	47	17	486	51	19
48	19	6	136	13	11	224	19	14	312	67	15	400	47	17	488	51	19
50	7	7	138	13	11	226	35	13	314	59	16	402	47	17	490	51	20
52	9	7	140	31	10	228	35	13	316	59	16	404	119	17	492	87	19
54	9	7	142	13	11	230	35	13	318	49	16	406	113	18	494	137	19
56	9	7	144	15	11	232	89	13	320	43	16	408	109	18	496	183	20
58	9	7	146	15	11	234	17	15	322	43	15	410	121	18	498	75	20
60	9	7	148	15	11	236	43	14	324	43	15	412	55	18	500	53	20
62	9	7	150	17	11	238	37	14	326	43	16	414	49	18	502	53	19
64	11	7	152	27	11	240	37	13	328	21	17	416	49	17	504	53	19
66	19	7	154	13	12	242	37	13	330	71	16	418	49	17	506	53	19
68	19	7	156	43	11	244	37	14	332	51	16	420	49	18	508	141	19
70	9	8	158	29	11	246	53	14	334	21	17	422	79	18	510	27	21
72	9	8	160	29	11	248	45	14	336	45	16	424	75	18	512	77	20
74	21	7	162	29	11	250	17	15	338	45	15	426	27	19	514	77	20
76	11	8	164	45	11	252	39	14	340	45	16	428	57	18	516	143	20
78	29	8	166	17	12	254	39	13	342	23	17	430	29	19	518	61	20
80	23	8	168	17	12	256	39	14	344	25	17	432	51	18	520	55	20
82	23	8	170	31	11	258	99	14	346	25	17	434	51	17	522	55	19
84	11	9	172	31	11	260	19	15	348	41	17	436	51	18	524	55	19
86	11	9	174	31	12	262	19	15	350	153	16	438	129	18	526	55	20
88	25	8	176	39	12	264	19	15	352	47	16	440	69	19	528	55	21

, we show the minimum diameter k and chord c for each number of vertices $N\le 528$ of a chordal ring graph $CR(N,k)$. The cases in which we get the maximum number of vertices for a given diameter are in boldface. It should be noted that, for an odd diameter k, such a maximum is as expected, whereas, for an even diameter k, the maximum supports Conjecture 1.

Other properties of the chordal ring graphs were studied by Barrière et al. [14] (gossiping), Barrière et al. [15] (fault-tolerant routing), and Zimmerman and Esfahanian [16] (fault tolerance).

Chordal Ring Graphs as Lifts

The chordal ring graph $CR(N,c)$, with $N=2n$, can be seen as a lift of a base graph on the group ${\mathbb{Z}}_n$, which is represented in Figure 1 (right). This allows us to derive a closed formula giving all the eigenvalues of $CR(N,c)$.

Proposition 2. 

Given integers $N=2n$ and c (odd), the eigenvalues of the chordal ring graph $CR(N,c)$ are

$$\lambda {\left(r\right)}_{1,2}=\pm \sqrt{4{cos}^2\left({\displaystyle \frac{r\pi }{n}}\right)+4{cos}^2\left({\displaystyle \frac{(c-1)r\pi }{2n}}\right)+4{cos}^2\left({\displaystyle \frac{(c+1)r\pi }{2n}}\right)-3}$$

(3)

for $r=0,1,\dots ,n-1$.

Proof. 

The polynomial matrix of the base graph in Figure 1 is

$$\mathit{B}\left(z\right)=\left(\begin{array}{cc}0& 1+{\displaystyle \frac{1}{z}}+z^{c^{\prime }}\\ 1+z+{\displaystyle \frac{1}{z^{c^{\prime }}}}& 0\end{array}\right),$$

where $c^{\prime }={\displaystyle \frac{c-1}{2}}$. Then, the eigenvalues of $CR(N,c)$ can be obtained as the eigenvalues of $\mathit{B}\left(z\right)$ for every $z={\zeta }^r$ with $\zeta =e^{{\displaystyle \frac{i2\pi }{n}}}$, for $n=N/2$ and $r=0,1,\dots ,n-1$. With $\tau =1+{\displaystyle \frac{1}{z}}+z^{c^{\prime }}$, such eigenvalues are

$$\begin{array}{cc}\hfill \lambda {\left(r\right)}_{1,2}& =\pm \sqrt{\tau \overline{\tau }}=\pm \left|\tau \right|\hfill \\ \hfill & =\pm \sqrt{{\left[1+cos\left({\displaystyle \frac{r2\pi }{n}}\right)+cos\left({\displaystyle \frac{c^{\prime }r2\pi }{n}}\right)\right]}^2+{\left[-sin\left({\displaystyle \frac{r2\pi }{n}}\right)+sin\left({\displaystyle \frac{c^{\prime }r2\pi }{n}}\right)\right]}^2}\hfill \end{array}$$

and, operating with $c^{\prime }={\displaystyle \frac{c-1}{2}}$, we obtain (3). □

Example 3. 

In the case of the graph $CR(20,5)$, we list its eigenvalues (3) for every $r=0,1,\dots ,9$ in

Table 1. All the eigenvalues of the matrices $\mathit{B}\left({\zeta }^r\right)$, which yield the eigenvalues of the chordal ring graph $CR(20,5)$.

r	$\mathit{\lambda }{\left(\mathit{r}\right)}_{1,2}$
0	$\pm 3$
1	$\pm \sqrt{3+2cos\left({\displaystyle \frac{\pi }{5}}\right)}\approx \pm 2.149$
2	$\pm 2cos\left({\displaystyle \frac{\pi }{5}}\right)-1\approx \pm 0.6164$
3	$\pm \sqrt{3-2cos\left({\displaystyle \frac{\pi }{5}}\right)}\approx \pm 1.543$
4	$\pm {\displaystyle \frac{1}{2}}(1+\sqrt{5})\approx \pm 1.618$
5	$\pm 1$
6	$\pm {\displaystyle \frac{1}{2}}(1+\sqrt{5})\approx \pm 1.618$
7	$\pm \sqrt{3+2cos\left({\displaystyle \frac{\pi }{5}}\right)}\approx \pm 1.543$
8	$\pm 2cos\left({\displaystyle \frac{\pi }{5}}\right)-1\approx \pm 0,6164$
9	$\pm \sqrt{3+2cos\left({\displaystyle \frac{\pi }{5}}\right)}\approx \pm 2.149$

.

3. Chordal Multi-Ring Graphs

To generalize chordal ring graphs, we introduce the family of chordal multi-ring graphs.

Definition 4. 

Given positive integers m, n (even), and $c(>1)$ (odd), the chordal m-ring graph $CMR(m,n,c)$ has vertices labeled with the elements of the Abelian group ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$, and edges $(\alpha ,i)\sim (\alpha ,i\pm 1)$ for every $\alpha \in {\mathbb{Z}}_m$ and $i\in {\mathbb{Z}}_n$, and $(\alpha ,i)\sim (\alpha +1,i+c)$ if i is even and $(\alpha ,i)\sim (\alpha -1,i-c)$ if i is odd.

Then, the graph $CMR(m,n,c)$ on $N=mn$ vertices is 3-regular and bipartite and consists of m cycles of even length $n=2\nu$, together with some edges joining them. In particular, when $m=1$, $CMR(1,n,c)\cong CR(n,c)$. For example, the chordal multi-ring graphs $CMR(2,12,3)$ and $CMR(3,18,3)$ are represented in Figure 6 (left and middle). As we show later, the first one is of special interest because all its eigenvalues are integers.

The adjacencies of the chordal m-ring graphs follow the same planar pattern for every $m\ge 1$; see Figure 5. Then, their maximum numbers of vertices are, again, those in (2). As we show in what follows, the advantage of using more than one cycle is that, for an even diameter k, the maximum number of vertices can be attained.

In

Table A2. The minimum diameter k in the chordal multi-ring graphs $CRM(m,n,c)$ with different values of m and n. In boldface are the cases with a maximum number of vertices for a given diameter.

N	m	n	c	k	N	m	n	c	k	N	m	n	c	k	N	m	n	c	k
4	1	4	3	2	46	1	46	7	7	80	1	80	23	8	108	9	12	3	10
6	1	6	3	2	48	1	48	19	6	80	2	40	5	8	110	1	110	11	10
8	1	8	3	3	48	2	24	5	7	80	4	20	3	8	110	5	22	3	10
8	2	4	3	3	48	3	16	3	6	80	5	16	5	9	112	1	112	25	9
10	1	10	3	3	48	4	12	5	7	80	8	10	3	9	112	2	56	7	10
12	1	12	5	3	48	6	8	3	7	82	1	82	23	8	112	4	28	3	10
12	2	6	5	3	50	1	50	7	7	84	1	84	11	9	112	7	16	3	10
12	3	4	3	3	50	5	10	3	7	84	2	42	5	9	112	8	14	3	10
14	1	14	5	3	52	1	52	9	7	84	3	28	3	9	114	1	114	25	9
16	1	16	5	4	52	2	26	5	7	84	6	14	3	9	114	3	38	17	9
16	2	8	3	4	54	1	54	9	7	84	7	12	3	9	116	1	116	13	10
16	4	4	3	4	54	3	18	3	6	86	1	86	11	9	116	2	58	11	10
18	1	18	5	4	56	1	56	9	7	88	1	88	25	8	118	1	118	15	10
18	3	6	3	4	56	2	28	5	7	88	2	44	5	10	120	1	120	27	10
20	1	20	5	4	56	4	14	5	7	88	4	22	3	8	120	2	60	7	11
20	2	10	3	4	56	7	8	3	7	90	1	90	11	9	120	3	40	5	10
22	1	22	5	5	58	1	58	9	7	90	3	30	3	9	120	4	30	11	10
24	1	24	5	5	60	1	60	9	7	90	5	18	5	9	120	5	24	3	10
24	2	12	3	4	60	2	30	5	7	90	9	10	3	9	120	6	20	5	11
24	3	8	3	5	60	3	20	3	7	92	1	92	11	9	120	10	12	3	11
24	4	6	3	5	60	5	12	3	7	92	2	46	7	9	122	1	122	27	9
26	1	26	5	5	60	6	10	3	7	94	1	94	11	9	124	1	124	27	10
28	1	28	7	5	62	1	62	9	7	96	1	96	11	9	124	2	62	19	10
28	2	14	3	5	64	1	64	11	7	96	2	48	7	9	126	1	126	13	11
30	1	30	7	5	64	2	32	5	8	96	3	32	7	9	126	3	42	5	10
30	3	10	5	5	64	4	16	3	8	96	4	24	3	8	126	7	18	3	11
30	5	6	3	5	64	8	8	3	8	96	6	16	7	9	126	9	14	3	11
32	1	32	7	5	66	1	66	19	7	96	8	12	5	9	128	1	128	13	11
32	2	16	3	6	66	3	22	11	7	98	1	98	11	9	128	2	64	13	10
32	4	8	3	5	68	1	68	19	7	98	7	14	3	9	128	4	32	3	12
34	1	34	7	5	68	2	34	11	7	100	1	100	13	9	128	8	16	3	11
36	1	36	7	6	70	1	70	9	8	100	2	50	7	9	130	1	130	29	10
36	2	18	3	6	70	5	14	5	8	100	5	20	3	10	130	5	26	3	10
36	3	12	3	6	70	7	10	3	8	100	10	10	3	10	132	1	132	29	10
36	6	6	3	6	72	1	72	9	8	102	1	102	13	9	132	2	66	13	10
38	1	38	15	5	72	2	36	5	8	102	3	34	7	9	132	3	44	5	10
40	1	40	7	6	72	3	24	3	9	104	1	104	23	9	132	6	22	5	11
40	2	20	3	6	72	4	18	3	8	104	2	52	7	9	132	11	12	3	11
40	4	10	3	6	72	6	12	3	8	104	4	26	3	9	134	1	134	13	11
40	5	8	3	6	74	1	74	21	7	106	1	106	23	9	136	1	136	13	11
42	1	42	7	6	76	1	76	11	8	108	1	108	41	9	136	2	68	7	11
42	3	14	3	6	76	2	38	5	8	108	2	54	7	9	136	4	34	13	11
44	1	44	9	6	78	1	78	29	8	108	3	36	5	10	138	1	138	13	11
44	2	22	7	6	78	3	26	13	8	108	6	18	5	10	138	3	46	5	11

, we show the minimum diameter k and chord c for each number of vertices $N\le 138$ of a chordal multi-ring graph.

Table A3. The minimum diameter k in the chordal multi-ring graphs $CMR(m,n,c)$ with minimum values m and n as in Table A2, but, now, with only one value of m and n. In boldface are the cases with a maximum number of vertices for a given diameter.

N	m	n	c	k	N	m	n	c	k	N	m	n	c	k	N	m	n	c	k
4	1	4	3	2	92	1	92	11	9	180	1	180	33	12	268	1	268	41	14
6	1	6	3	2	94	1	94	11	9	182	1	182	33	11	270	1	270	21	15
8	1	8	3	3	96	4	24	3	8	184	1	184	33	12	272	1	272	23	15
10	1	10	3	3	98	1	98	11	9	186	1	186	51	12	274	1	274	37	15
12	1	12	5	3	100	1	100	13	9	188	1	188	15	13	276	1	276	37	15
14	1	14	5	3	102	1	102	13	9	190	1	190	79	12	278	1	278	37	15
16	1	16	5	4	104	1	104	23	9	192	1	192	35	12	280	1	280	43	14
18	1	18	5	4	106	1	106	23	9	194	1	194	35	12	282	1	282	37	15
20	1	20	5	4	108	1	108	41	9	196	1	196	17	13	284	1	284	61	15
22	1	22	5	5	110	1	110	11	10	198	1	198	17	13	286	1	286	21	16
24	2	12	3	4	112	1	112	25	9	200	1	200	17	13	288	1	288	77	15
26	1	26	5	5	114	1	114	25	9	202	1	202	17	13	290	1	290	39	15
28	1	28	7	5	116	1	116	13	10	204	1	204	37	12	292	1	292	39	15
30	1	30	7	5	118	1	118	15	10	206	1	206	19	13	294	7	42	3	14
32	1	32	7	5	120	1	120	27	10	208	1	208	57	13	296	1	296	39	15
34	1	34	7	5	122	1	122	27	9	210	1	210	77	13	298	1	298	39	16
36	1	36	7	6	124	1	124	27	10	212	1	212	33	13	300	1	300	69	15
38	1	38	15	5	126	3	42	5	10	214	1	214	33	13	302	1	302	65	15
40	1	40	7	6	128	2	64	13	10	216	6	36	3	12	304	1	304	41	16
42	1	42	7	6	130	1	130	29	10	218	1	218	33	13	306	1	306	41	15
44	1	44	9	6	132	1	132	29	10	220	1	220	17	14	308	1	308	41	15
46	1	46	7	7	134	1	134	13	11	222	1	222	51	13	310	1	310	41	15
48	1	48	19	6	136	1	136	13	11	224	4	56	5	13	312	1	312	67	15
50	1	50	7	7	138	1	138	13	11	226	1	226	35	13	314	1	314	59	16
52	1	52	9	7	140	1	140	31	10	228	1	228	35	13	316	1	316	59	16
54	3	18	3	6	142	1	142	13	11	230	1	230	35	13	318	1	318	49	16
56	1	56	9	7	144	1	144	15	11	232	1	232	89	13	320	1	320	43	16
58	1	58	9	7	146	1	146	15	11	234	3	78	7	14	322	1	322	43	15
60	1	60	9	7	148	1	148	15	11	236	1	236	43	14	324	1	324	43	15
62	1	62	9	7	150	5	30	3	10	238	1	238	37	14	326	1	326	43	16
64	1	64	11	7	152	1	152	27	11	240	1	240	37	13	328	2	164	23	16
66	1	66	19	7	154	1	154	13	12	242	1	242	37	13	330	1	330	71	16
68	1	68	19	7	156	1	156	43	11	244	1	244	37	14	332	1	332	51	16
70	1	70	9	8	158	1	158	29	11	246	1	246	53	14	334	1	334	21	17
72	1	72	9	8	160	1	160	29	11	248	1	248	45	14	336	1	336	45	16
74	1	74	21	7	162	1	162	29	11	250	1	250	17	15	338	1	338	45	15
76	1	76	11	8	164	1	164	45	11	252	1	252	39	14	340	1	340	45	16
78	1	78	29	8	166	1	166	17	12	254	1	254	39	13	342	3	114	15	16
80	1	80	23	8	168	1	168	17	12	256	1	256	39	14	344	1	344	25	17
82	1	82	23	8	170	1	170	31	11	258	1	258	99	14	346	1	346	25	17
84	1	84	11	9	172	1	172	31	11	260	1	260	19	15	348	1	348	41	17
86	1	86	11	9	174	1	174	31	12	262	1	262	19	15	350	1	350	153	16
88	1	88	25	8	176	1	176	39	12	264	2	132	19	14	352	1	352	47	16
90	1	90	11	9	178	1	178	49	12	266	1	266	41	14	354	1	354	47	16

provides the same results, but, now, with only one value of $mn$ up to $N\le 354$. The cases in which we get the maximum number of vertices for a given diameter are in boldface. As commented above, we observe that, for an even diameter, the number of vertices of the chordal multi-ring graphs attains the maximum possible value. This can be proved in general.

Proposition 5. 

For an even diameter $k\ge 2$, the chordal m-ring graph $CMR(m,n,c)$ with $m=k/2$, $n=3k$, and $c=3$ has the maximum possible order $N={\displaystyle \frac{3}{2}}{k}^2$.

Proof. 

For every even k, an optimal tile with an area $3k^2/2$ periodically tessellates the plane and corresponds to a chordal m-ring graph with $m=k/2$. See Figure 7 (up) for $k=2,4,6,8$, where the distances from vertex 0 are indicated. Then, from the distribution of the 0s, it can be checked that the corresponding graph has chord $c=3$. See Figure 7 (down) for $k=6$ with the vertices labeled like in Definition 4. The obtained chordal 3-ring graph is shown in Figure 6 (middle). □

Chordal multi-ring graphs can also be represented as lift graphs; see their base graph in Figure 6 (right). Then, we have the following result, which gives the eigenvalues of a chordal m-ring graph $CMR(m,n,c)$.

Proposition 6. 

Given integers m, $n=2\nu$ (even), and c (odd), the eigenvalues of the chordal m-ring graph $CMR(m,n,c)$ are

$$\begin{array}{cc}\hfill \lambda {(r,s)}_{1,2}& =\\ & \pm \sqrt{4{cos}^2\left({\displaystyle \frac{s\pi }{\nu }}\right)+4{cos}^2\left({\displaystyle \frac{r\pi }{m}}+{\displaystyle \frac{(c-1)s\pi }{n}}\right)+4{cos}^2\left({\displaystyle \frac{r\pi }{m}}+{\displaystyle \frac{(c+1)s\pi }{n}}\right)-3}\hfill \end{array}$$

(4)

for $r=0,1,\dots ,m-1$ and $s=0,1,\dots ,\nu -1$.

Proof. 

The base graph of $CMR(m,n,c)$ is shown in Figure 6 (right). Then, its $(y,z)$-polynomial matrix is

$$\mathit{B}(y,z)=\left(\begin{array}{cc}0& 1+{\displaystyle \frac{1}{z}}+y{z}^{c^{\prime }}\\ 1+z+{\displaystyle \frac{1}{y{z}^{c^{\prime }}}}& 0\end{array}\right),$$

where $c^{\prime }={\displaystyle \frac{c-1}{2}}$. Then, the eigenvalues of $CMR(m,n,c)$ can be obtained as the eigenvalues of $\mathit{B}(y,z)$ for every $y=e^{{\displaystyle \frac{i2\pi }{m}}r}$ with $\nu =n/2$, $r=0,\dots ,m-1$, and $z=e^{{\displaystyle \frac{i2\pi }{\nu }}s}$ for $s=0,\dots ,\nu -1$. With $\tau =1+{\displaystyle \frac{1}{z}}+y{z}^{c^{\prime }}$, such eigenvalues are

$$\begin{array}{cc}\hfill & \lambda {\left(r\right)}_{1,2}=\pm \sqrt{\tau \overline{\tau }}=\pm \left|\tau \right|\hfill \\ \hfill & =\pm \sqrt{{\left[1+cos\left({\displaystyle \frac{s2\pi }{\nu }}\right)+cos\left({\displaystyle \frac{r2\pi }{m}}+{\displaystyle \frac{c^{\prime }s2\pi }{\nu }}\right)\right]}^2+{\left[-sin\left({\displaystyle \frac{s2\pi }{\nu }}\right)+sin\left({\displaystyle \frac{r2\pi }{m}}+{\displaystyle \frac{c^{\prime }s2\pi }{\nu }}\right)\right]}^2}\hfill \end{array}$$

and, operating with $c^{\prime }={\displaystyle \frac{c-1}{2}}$, we obtain (4). □

Figure 6 shows the chordal multi-ring graphs $CMR(2,12,3)$ and $CMR(3,18,3)$. Their eigenvalues are shown in

Table 2. The eigenvalues of the chordal multi-ring graph $CMR(2,12,3)$.

$\mathit{r}\setminus \mathit{s}$	0	1	2	3	4	5
0	$\pm 3$	$\pm 2$	$\pm 0$	$\pm 1$	$\pm 0$	$\pm 2$
1	$\pm 1$	$\pm 2$	$\pm 2$	$\pm 1$	$\pm 2$	$\pm 2$

and

Table 3. The eigenvalues of the chordal multi-ring graph $CMR(3,18,3)$.

$\mathit{r}\setminus \mathit{s}$	0	1	2	3	4	5	6	7	8
0	$\pm 3$	$\pm 0.8794$	$\pm 2.532$	$\pm 0$	$\pm 1.348$	$\pm 1.348$	$\pm 0$	$\pm 2.532$	$\pm 0.8794$
1	$\pm 1.732$	$\pm 2.532$	$\pm 1.348$	$\pm 1.732$	$\pm 0.8794$	$\pm 0.8794$	$\pm 1.732$	$\pm 1.348$	$\pm 2.532$
2	$\pm 1.732$	$\pm 2.532$	$\pm 1.348$	$\pm 1.732$	$\pm 0.8794$	$\pm 0.8794$	$\pm 1.732$	$\pm 1.348$	$\pm 2.532$

with $y=e^{ri{\displaystyle \frac{2\pi }{2}}}$ and $z=e^{si{\displaystyle \frac{2\pi }{6}}}$. It should be noted that the graph $CMR(2,12,3)$ has an integral spectrum; see, for example, Ahmadi et al. [17]. As commented in that paper, such graphs play an important role in quantum networks supporting the so-called perfect state transfer.

4. Chordal Ring Mixed Graphs

In a mixed graph, there are edges (without direction) and arcs (with direction). See some recent results of some mixed graphs in Dalfó et al. [18] and Dalfó et al. [7].

Definition 7. 

Let $N\ge 2$ and $c(<N)$ be, respectively, even and odd numbers. The chordal ring mixed graph $CRM(N,c)$ is a mixed graph with vertex set $V={\mathbb{Z}}_N$ (all arithmetic is modulo N), with arcs $i\to i+1$ (forming a directed cycle) and edges $i\sim i+c$ if i is even (and, hence, $i\sim i-c$ if i is odd, forming the ‘chords’).

See the example of the chordal ring mixed graph $CRM(10,3)$ in Figure 8 (left).

As in the case of chordal ring graphs $CR(N,c)$, if each vertex of $CRM(N,c)$ is represented by a numbered unit square modulo N, the vertices reached at a distance $0,1,2,\dots$ from any given vertex can be arranged in a planar pattern, as shown in Figure 9 (starting from vertex 0).

Since there are at most $\ell +1$ vertices at a distance $\ell (>0)$ from vertex 0, and the graph is bipartite; the maximum number N of vertices of a chordal ring mixed graph with a diameter k is

$$N\left(k\right)=\left\{\begin{array}{cc}{\displaystyle \frac{1}{2}}{(k+1)}^2\hfill & \mathrm{for} k \mathrm{odd},\hfill \\ {\displaystyle \frac{1}{2}}k(k+2)\hfill & \mathrm{for} k \mathrm{even}.\hfill \end{array}\right.$$

(5)

Moreover, Dalfó et al. [18] showed that such a maximum can be attained when k is odd but cannot be attained when $k(>2)$ is even. We can raise the following conjecture from the reasoning in [18] and computer exploration shown in

Table A4. The minimum diameter k for each number of vertices $N\le 526$ in the chordal ring mixed graphs $CRM(N,c)$ with (minimum) chord c. In boldface are the cases with a maximum number of vertices for a given diameter.

N	c	k	N	c	k	N	c	k	N	c	k	N	c	k	N	c	k
2	1	1	90	33	13	178	17	19	266	23	23	354	75	27	442	139	29
4	1	3	92	11	15	180	17	19	268	61	23	356	23	27	444	27	31
6	3	3	94	11	13	182	19	19	270	97	23	358	23	27	446	27	29
8	3	3	96	9	15	184	51	19	272	41	23	360	165	27	448	23	31
10	3	4	98	13	13	186	33	19	274	43	23	362	25	27	450	29	29
12	3	5	100	13	15	188	15	19	276	19	23	364	25	27	452	21	31
14	3	5	102	39	15	190	41	19	278	65	23	366	27	27	454	81	31
16	3	6	104	9	15	192	69	19	280	87	23	368	21	27	456	99	31
18	5	5	106	11	15	194	85	19	282	127	23	370	115	27	458	53	31
20	3	7	108	33	15	196	17	19	284	21	23	372	87	27	460	55	31
22	5	7	110	13	15	198	17	21	286	21	25	374	49	27	462	25	31
24	5	7	112	13	15	200	19	19	288	23	23	376	51	27	464	25	31
26	7	7	114	15	15	202	91	20	290	133	24	378	57	27	466	61	31
28	5	7	116	11	15	204	15	21	292	81	25	380	23	27	468	99	31
30	5	9	118	27	15	206	31	21	294	19	25	382	87	27	470	89	31
32	7	7	120	33	15	208	37	21	296	19	25	384	117	27	472	23	31
34	13	8	122	51	15	210	33	21	298	17	25	386	177	27	474	27	31
36	15	9	124	13	15	212	17	21	300	47	25	388	25	27	476	133	31
38	5	9	126	11	17	214	17	21	302	41	25	390	19	29	478	29	31
40	7	9	128	15	15	216	99	21	304	21	25	392	27	27	480	29	31
42	9	9	130	57	16	218	15	21	306	21	25	394	183	28	482	31	31
44	13	10	132	39	17	220	19	21	308	143	25	396	75	29	484	105	31
46	7	9	134	25	17	222	21	21	310	23	25	398	71	29	486	201	31
48	5	11	136	13	17	224	51	22	312	23	25	400	91	29	488	25	31
50	9	9	138	11	17	226	69	21	314	19	25	402	23	29	490	145	31
52	7	11	140	63	17	228	13	23	316	85	26	404	23	29	492	57	31
54	7	11	142	15	17	230	17	21	318	69	25	406	47	29	494	59	31
56	21	11	144	15	17	232	15	23	320	37	27	408	93	29	496	65	31
58	9	11	146	17	17	234	69	21	322	133	25	410	21	29	498	87	31
60	7	11	148	41	18	236	19	23	324	21	27	412	25	29	500	27	31
62	11	11	150	13	17	238	19	21	326	21	25	414	25	29	502	119	31
64	19	11	152	13	19	240	19	23	328	21	27	416	195	29	504	177	31
66	25	11	154	47	17	242	21	21	330	117	25	418	27	29	506	235	31
68	9	11	156	15	19	244	21	23	332	17	27	420	27	29	508	29	31
70	9	13	158	15	17	246	17	23	334	23	25	422	29	29	510	29	33
72	11	11	160	11	19	248	15	23	336	23	27	424	99	30	512	31	31
74	31	12	162	17	17	250	67	23	338	25	25	426	23	29	512	31	31
76	9	13	164	13	19	252	47	23	340	25	27	428	19	31	514	241	32
78	9	13	166	31	19	254	45	23	342	123	27	430	91	29	516	135	33
80	35	13	168	45	19	256	19	23	344	45	27	432	21	31	518	23	33
82	11	13	170	39	19	258	19	23	346	21	27	434	189	29	520	165	33
84	11	13	172	15	19	260	55	23	348	21	27	436	25	31	522	93	33
86	9	13	174	15	19	262	21	23	350	55	27	438	25	29	524	71	33
88	19	14	176	13	19	264	17	23	352	19	27	440	25	31	526	27	33

.

Conjecture 2. 

The maximum number N of vertices of a chordal ring mixed graph $CRM(N,c)$ with an even diameter $k>2$ is $N={\displaystyle \frac{1}{2}}{k}^2+2$ if $k\equiv 0mod4$ (with $c={\displaystyle \frac{1}{4}}{n}^2-{\displaystyle \frac{1}{2}}n+1$), and $N=k({\displaystyle \frac{k}{2}}-1)+4$ if $k\equiv 2mod4$.

As in the case of chordal ring graphs, the method used in [18] to find the optimal ring mixed graphs consists of the same steps (1) and (2) from the introduction. For example, Figure 3 (right) shows the optimal tile for a diameter $k=5$ and its periodic tessellation.

In Table A4, we show the minimum diameter k and chord c for each number of vertices $N\le 526$ of a chordal ring mixed graph. The cases in which we get the maximum number of vertices for a given diameter are in boldface.

Chordal Ring Mixed Graphs as Lifts

The chordal ring mixed graph $CRM(N,c)$ can be seen as a lift of a base mixed graph on the group ${\mathbb{Z}}_{N/2}$, which is represented in Figure 8 (right).

Proposition 8. 

Given integers $N=2n$ and c (odd), the eigenvalues of the chordal ring graph mixed graphs $CRM(N,c)$ are

$$\pm {\lambda }_{1,2}\left(r\right)=\sqrt{z^{c^{\prime }}(z^{1+2c^{\prime }}+z^{1+c^{\prime }}+z^{c^{\prime }}+1})z^{-c^{\prime }},$$

(6)

where $z=e^{{\displaystyle \frac{i2\pi }{n}}r}$, $c^{\prime }={\displaystyle \frac{c-1}{2}}$, and $r=0,1,\dots ,n-1$.

Proof. 

First, it should be noted that $CRM(N,c)$ can be obtained as the lift of the base graph on the right side of Figure 8. The polynomial matrix $\mathit{B}\left(z\right)$, with $z=e^{{\displaystyle \frac{i2\pi }{n}}r}$, of such a base graph is

$$\mathit{B}\left(z\right)=\left(\begin{array}{cc}0& 1+z^{c^{\prime }}\\ z+{\displaystyle \frac{1}{z^{c^{\prime }}}}& 0\end{array}\right),$$

where $z=e^{{\displaystyle \frac{i2\pi }{n}}r}$ and $c^{\prime }=(c-1)/2$. Then, the eigenvalues of the lift $CRM(n,c)$ are the eigenvalues of $\mathit{B}\left(z\right)$ for $r=0,1,\dots ,n-1$, given in (6). □

It should be noted that, in general, the matrix $\mathit{B}\left(z\right)$ is not Hermitian; hence, the obtained eigenvalues are complex numbers.

Example 9. 

We consider the case of $CRM(20,5)$, that is, the chordal ring mixed graph with 20 vertices and, for each vertex i, there are edges $i\sim i\pm 5$ and one arc $i\to i+1$ (see Definition 7). We obtain the eigenvalues in

Table 4. All the eigenvalues of the chordal ring mixed graph $CRM(20,5)$.

$\mathit{\zeta }={\mathit{e}}^{\mathit{i}\frac{2\mathit{\pi }}{10}}$, $\mathit{z}={\mathit{\zeta }}^{\mathit{r}}$	$\mathit{\lambda }$
$sp\left(\mathit{B}\right(1\left)\right)$	$\pm 2$
$sp\left(\mathit{B}\right(\zeta \left)\right)$	$\pm (1.362+0.2158i)$
$sp\left(\mathit{B}\left({\zeta }^2\right)\right)$	$\pm (0.1913-0.5879i)$
$sp\left(\mathit{B}\left({\zeta }^3\right)\right)$	$\pm (0.9661+0.4922i)$
$sp\left(\mathit{B}\left({\zeta }^4\right)\right)$	$\pm (1.309+0.9511i)$
$sp\left(\mathit{B}\left({\zeta }^5\right)\right)$	$\pm 0$
$sp\left(\mathit{B}\left({\zeta }^6\right)\right)$	$\pm (1.309-0.9511i)$
$sp\left(\mathit{B}\left({\zeta }^7\right)\right)$	$\pm (0.9661-0.4922i)$
$sp\left(\mathit{B}\left({\zeta }^8\right)\right)$	$\pm (0.1913+0.5879i)$
$sp\left(\mathit{B}\left({\zeta }^9\right)\right)$	$\pm (1.362-0.2158i)$

, which are represented in Figure 10.

5. Chordal Multi-Ring Mixed Graphs

As in the case of chordal ring graphs, we can consider the mixed version of the chordal multi-ring graphs, denoted $CMRM(m,n,c)$, which, as mixed graphs, have edges and arcs.

Definition 10. 

Given positive integers m, n, (even) and $c(>1)$ (odd), the chordal m-ring mixed graph $CMRM(m,n,c)$ has vertices labeled with the elements of the Abelian group ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$, arcs $(\alpha ,i)\to (\alpha ,i+1)$ for every $\alpha \in {\mathbb{Z}}_m$ and $i\in {\mathbb{Z}}_n$, and edges $(\alpha ,i)\sim (\alpha +1,i+c)$ if i is even and $(\alpha ,i)\sim (\alpha -1,i-c)$ if i odd.

See the example of the chordal multi-ring mixed graph $CMRM(2,10,3)$ in Figure 11 (on the left), and its eigenvalues in

Table 5. All the eigenvalues of the chordal multi-ring mixed graph $CMRM(2,10,3)$.

$\mathit{s}\setminus \mathit{r}$	0	1
0	$\pm 2$	$\pm 0$
1	$\pm (0.9511+0.3090i)$	$\pm (0.8419+i0.1337)$
2	$\pm (1.733+0.2744i)$	$\pm (1.422+i0.4620)$
3	$\pm (0.3870-0.75941i)$	$\pm (1.563+i0.7965)$
4	$\pm (-0.5878+0.8090i)$	$\pm (1.210+i0.8790)$

.

Similar to the case of chordal multi-ring graphs, the adjacencies of the chordal m-ring mixed graphs follow the same planar pattern for every $m\ge 1$; see Figure 9. Then, their maximum numbers of vertices are, again, those in (5). Now, by using more than one cycle, we can improve the number of vertices reached for an even diameter k, but, in this case, without getting the possible maximum value in (5). More precisely, as shown in Conjecture 3 below, we get $N={\displaystyle \frac{1}{2}}{k}^2+2$ when $k\equiv 2mod4$ (instead of $N=k({\displaystyle \frac{k}{2}}-1)+4$ in Conjecture 2).

In

Table A5. The minimum diameter k in the chordal multi-ring mixed graphs $CMRM(m,n,c)$ with minimum values of m and n. In boldface are the cases with a maximum number of vertices for a given diameter.

N	m	n	c	k	N	m	n	c	k	N	m	n	c	k	N	m	n	c	k
4	1	4	3	3	92	2	46	5	13	180	1	180	17	19	268	1	268	61	23
6	1	6	3	3	94	1	94	11	13	182	1	182	19	19	270	1	270	97	23
8	1	8	3	3	96	1	96	9	15	184	1	184	51	19	272	1	272	41	23
10	1	10	3	4	98	1	98	13	13	186	1	186	33	19	274	1	274	43	23
12	1	12	3	5	100	2	50	7	14	188	1	188	15	19	276	1	276	19	23
14	1	14	3	5	102	1	102	39	15	190	1	190	41	19	278	1	278	65	23
16	1	16	3	6	104	1	104	9	15	192	1	192	69	19	280	1	280	87	23
18	1	18	5	5	106	1	106	11	15	194	1	194	85	19	282	1	282	127	23
20	2	10	3	6	108	1	108	33	15	196	1	196	17	19	284	1	284	21	23
22	1	22	5	7	110	1	110	13	15	198	9	22	11	19	286	11	26	13	23
24	1	24	5	7	112	1	112	13	15	200	1	200	19	19	288	1	288	23	23
26	1	26	7	7	114	1	114	15	15	202	1	202	91	20	290	1	290	133	24
28	1	28	5	7	116	1	116	11	15	204	1	204	15	21	292	1	292	81	25
30	3	10	5	7	118	1	118	27	15	206	1	206	31	21	294	1	294	19	25
32	1	32	7	7	120	1	120	33	15	208	1	208	37	21	296	1	296	19	25
34	1	34	13	8	122	1	122	51	15	210	1	210	33	21	298	1	298	17	25
36	1	36	15	9	124	1	124	13	15	212	1	212	17	21	300	1	300	47	25
38	1	38	5	9	126	7	18	9	15	214	1	214	17	21	302	1	302	41	25
40	1	40	7	9	128	1	128	15	15	216	1	216	99	21	304	1	304	21	25
42	1	42	9	9	130	1	130	57	16	218	1	218	15	21	306	1	306	21	25
44	2	22	3	9	132	1	132	39	17	220	1	220	19	21	308	1	308	143	25
46	1	46	7	9	134	1	134	25	17	222	1	222	21	21	310	1	310	23	25
48	1	48	5	11	136	1	136	13	17	224	1	224	51	22	312	1	312	23	25
50	1	50	9	9	138	1	138	11	17	226	1	226	69	21	314	1	314	19	25
52	2	26	5	10	140	1	140	63	17	228	2	114	33	21	316	2	158	33	25
54	1	54	7	11	142	1	142	15	17	230	1	230	17	21	318	1	318	69	25
56	1	56	21	11	144	1	144	15	17	232	4	58	25	21	320	1	320	37	27
58	1	58	9	11	146	1	146	17	17	234	1	234	69	21	322	1	322	133	25
60	1	60	7	11	148	2	74	23	17	236	2	118	9	21	324	2	162	57	25
62	1	62	11	11	150	1	150	13	17	238	1	238	19	21	326	1	326	21	25
64	1	64	19	11	152	4	38	13	17	240	1	240	19	23	328	4	82	33	25
66	1	66	25	11	154	1	154	47	17	242	1	242	21	21	330	1	330	117	25
68	1	68	9	11	156	2	78	7	17	244	2	122	11	22	332	2	166	11	25
70	5	14	7	11	158	1	158	15	17	246	1	246	17	23	334	1	334	23	25
72	1	72	11	11	160	1	160	11	19	248	1	248	15	23	336	1	336	23	27
74	1	74	31	12	162	1	162	17	17	250	1	250	67	23	338	1	338	25	25
76	1	76	9	13	164	2	82	9	18	252	1	252	47	23	340	2	170	13	26
78	1	78	9	13	166	1	166	31	19	254	1	254	45	23	342	1	342	123	27
80	1	80	35	13	168	1	168	45	19	256	1	256	19	23	344	1	344	45	27
82	1	82	11	13	170	1	170	39	19	258	1	258	19	23	346	1	346	21	27
84	1	84	11	13	172	1	172	15	19	260	1	260	55	23	348	1	348	21	27
86	1	86	9	13	174	1	174	15	19	262	1	262	21	23	350	1	350	55	27
88	4	22	9	13	176	1	176	13	19	264	1	264	17	23	352	1	352	19	27
90	1	90	33	13	178	1	178	17	19	266	1	266	23	23	354	1	354	75	27

, we show the minimum diameter k and chord c for each number of vertices $N\le 354$ of a chordal multi-ring mixed graph. The cases in which we get the maximum number of vertices for a given diameter are in boldface. From these values, we pose the following conjecture.

Conjecture 3. 

The maximum number N of vertices of a chordal m-ring mixed graph with an even diameter $k\equiv 2mod4$ is $N={\displaystyle \frac{1}{2}}{k}^2+2$, and this value is attained with $m=2$ cycles and chord $c=k/2$.

We have the following result concerning the spectrum of the chordal multi-ring mixed graphs.

Proposition 11. 

Given integers m, n (even), and c (odd), the eigenvalues of the chordal multi-ring mixed graph $CMRM(m,n,c)$ are

$$\pm {\lambda }_{1,2}\left(r\right)=\sqrt{y{z}^{c^{\prime }}(y^2{z}^{1+2c^{\prime }}+y{z}^{1+c^{\prime }}+y{z}^{c^{\prime }}+1)}{y}^{-1}{z}^{-c^{\prime }},$$

(7)

where $y=e^{{\displaystyle \frac{i2\pi }{m}}r}$, for $r=0,1,\dots ,m-1$, $z=e^{{\displaystyle \frac{i2\pi }{n}}s}$, $s=0,1,\dots ,n-1$, and $c^{\prime }={\displaystyle \frac{c-1}{2}}$.

Proof. 

In this case, $CMRM(m,n,c)$ can be obtained as the lift of the base graph on the right side of Figure 11. The polynomial matrix $\mathit{B}(y,z)$ of such a base graph is

$$\mathit{B}(y,z)=\left(\begin{array}{cc}0& 1+y{z}^{c^{\prime }}\\ z+{\displaystyle \frac{1}{y{z}^{c^{\prime }}}}& 0\end{array}\right),$$

with $y=e^{{\displaystyle \frac{i2\pi }{m}}r}$ and $z=e^{{\displaystyle \frac{i2\pi }{n}}r}$. Then, the eigenvalues of the lift $CMRM(m,n,c)$ are the eigenvalues of $\mathit{B}(y,z)$ for $r=0,1,\dots ,n-1$, and $s=0,1,\dots ,n-1$ given in (7). □

6. Discussion

In this paper, we have obtained the following results:

• We have introduced the families of chordal multi-ring (CMR), chordal ring mixed (CMR), and chordal multi-ring mixed (CMRM) graphs from the well-known chordal ring (CR) graphs.
• For these four families, we have obtained the maximum number of vertices for every given diameter by using the approach of plane tessellations and their corresponding lattices. Moreover, we computed the minimum diameter for any value of the number of vertices. Actually, for our families of graphs, these are optimization results that solve the degree/diameter problem and the degree/number of vertices problem, which are very well-known problems in the literature. See, for example, the comprehensive survey by Miller and Širáň [19].
• Finally, we have obtained the spectra of these four families. Here, we used the theory of lifts of some base graphs on Abelian groups to derive closed formulas for the eigenvalues of our families of graphs.