Largest regular multigraphs with three distinct eigenvalues

TL;DR

This paper investigates the maximum size of connected regular multigraphs with exactly three distinct eigenvalues, establishing bounds and conditions for their existence based on algebraic and combinatorial properties.

Contribution

It provides an upper bound on the number of vertices for such graphs and characterizes cases of equality linked to finite projective planes and polarities.

Findings

For most degrees k, the maximum number of vertices is bounded by k^2 - k + 1.

Moore graphs are the largest for k=2,3,7.

Existence of certain graphs is tied to finite projective planes with polarities.

Abstract

We deal with connected $k$-regular multigraphs of order $n$ that has only three distinct eigenvalues. In this paper, we study the largest possible number of vertices of such a graph for given $k$. For $k=2,3,7$, the Moore graphs are largest. For $k\ne 2,3,7,57$, we show an upper bound $n\leq k^2-k+1$, with equality if and only if there exists a finite projective plane of order $k-1$ that admits a polarity.

Equations

$$F_0^{(k)}(x)=1,F_1^{(k)}(x)=x,F_2^{(k)}(x)=x^2-k,$$

$$F_i^{(k)}(x)=x{F}_{i-1}^{(k)}(x)-(k-1)F_{i-2}^{(k)}(x)$$

$$f_{uv}^{(i)}$$

$$=\sum _{s\in V}{b}_{us}^{(1)}{b}_{sv}^{(i-1)}-(k-1)b_{uv}^{(i-2)}.$$

$$n\le \frac{f(k)}{f_0}.$$

$$\sum _{j=0}^{d}f({\tau }_j){\mathbf{E}}_j=f(\mathbf{A})=\sum _{i=0}^s{f}_i{F}_i^{(k)}(\mathbf{A}).$$

$$\begin{gathered} f(k)=\mathrm{tr}(f(k){\mathbf{E}}_0)\ge \mathrm{tr}\left(\sum _{j=0}^{d}f({\tau }_j){\mathbf{E}}_j\right) \\ =\mathrm{tr}\left(\sum _{i=0}^s{f}_i{F}_i^{(k)}(\mathbf{A})\right)\ge \mathrm{tr}(f_0\mathbf{I})=n{f}_0, \end{gathered}$$

$$F(x)=\sum _{i=0}^s{f}_i{F}_i^{(k)}(x)$$

$$n\le \sum _{i\in \{0,\dots ,d\}:f_i>0}{k}_i.$$

$$n^2\le \sum _{i\in \{0,\dots ,s\}:f_i>0}n{k}_i.$$

$$H_G(x)=n\prod _{i=1}^d\frac{x-{\tau }_i}{k-{\tau }_i}.$$

$$F_G(x)=F_2^{(k)}(x)-({\tau }_1+{\tau }_2)F_1^{(k)}(x)+(k+{\tau }_1{\tau }_2)F_0^{(k)}(x).$$

$$|\{w\in V:\partial (u,w)\le 2\}|$$

$$\le 1+(k-1)+|\{w\in V:\partial (u,w)=2,(v,w)\in E\}|$$

$$+|\{w\in V:\partial (u,w)=2,(v,w)\notin E\}|$$

$$\le 1+(k-1)+(k-2)+(k-1)(k-2)=k^2-k,$$

$$H_G(x)=n\frac{(x-{\tau }_1)(x-{\tau }_2)}{(k-{\tau }_1)(k-{\tau }_2)}.$$

$$n({\mathbf{A}}^2-({\tau }_1+{\tau }_2)\mathbf{A}+{\tau }_1{\tau }_2\mathbf{I})=(k-{\tau }_1)(k-{\tau }_2)\mathbf{J},$$

$$l_v^2-({\tau }_1+{\tau }_2+1)l_v=\frac{1}{n}(k-{\tau }_1)(k-{\tau }_2)-k-{\tau }_1{\tau }_2.$$

$$n\le 1+(k-2)+(k-2)(k-2)=k^2-3k+3<k^2-k+1,$$

$$n=k+1+\frac{k^2-\lambda k-k}{\mu }\le \frac{k^2}{2}+\frac{k}{2}+1\le k^2-k+1$$

$$n=1+k+k^2-s(\lambda +1{)}^2\le 1+k+k^2-3k<k^2-k+1.\text{qed}$$

$$\left(\begin{array}{cc}O& \mathbf{A}\\ {\mathbf{A}}^{\top }& O\end{array}\right),$$

$$n\le 1+(k-2)+(k-2)(k-1)=k^2-2k+1<k^2-k+1.$$

Full text

0002010 Mathematics Subject Classification: 05C50 (05D05)

Hiroshi Nozaki: Department of Mathematics Education, Aichi University of Education, 1 Hirosawa, Igaya-cho, Kariya, Aichi 448-8542, Japan. hnozaki@auecc.aichi-edu.ac.jp.

Key words: Graph spectrum, Moore bound, linear programming bound, projective plane,

1 Introduction

Let $G$ be a connected $k$-regular multigraph $(V,E)$, which may have a loop. For $u,v\in V$, let $m(u,v)$ be the number of edges between $u$ and $v$ if $u\neq v$, and the number of loops on $u$ if $u=v$. The adjacency matrix $\boldsymbol{A}$ of $G$ is defined to be the square matrix indexed by $V$ whose $(u,v)$ entry is $m(u,v)$ if $\{u,v\}\in E$ and [math] otherwise. The eigenvalues of $\boldsymbol{A}$ are called the eigenvalues of $G$. In this paper, we deal with a $k$-regular multigraph $G$ with only 3 distinct eigenvalues. Since the degree of the minimal polynomial of $\boldsymbol{A}$ is $3$, the diameter of $G$ is at most $2$. This implies that the Moore bound $|V|\leq k^{2}+1$ holds for $k$-regular multigraphs with only 3 distinct eigenvalues. If $G$ attains this bound, $G$ is called a Moore graph, which is simple. A Moore graph does not exist except for $(d,k)=(2,2),(2,3),(2,7),(2,57)$ [2, 5]. The following Moore graphs uniquely exist: the 5-cycle for $k=2$, the Petersen graph for $k=3$, and the Hoffman–Singleton graph for $k=7$ [9]. For $k=57$, the existence of the Moore graph is still open. The main problem of this paper is to improve the Moore bound, and to determine the largest $k$-regular multigraph with only 3 distinct eigenvalues for given $k\geq 3$.

A $k$-regular simple graph of order $n$ is called a strongly regular graph with parameters $(n,k,\lambda,\mu)$ if there exist integers $\lambda$ and $\mu$ such that any two adjacent vertices have $\lambda$ common neighbours, and any two non-adjacent vertices have $\mu$ common neighbours. If a connected regular simple graph has only 3 distinct eigenvalues, then it is strongly regular. If a connected $k$-regular simple graph satisfies that any two adjacent vertices have at least $\lambda$ common neighbours, and any two non-adjacent vertices have at least $\mu$ common neighbours, then the order $n$ has the bound $n\leq k+1+k(k-1-\lambda)/\mu$ (see [3]). Strongly regular graphs are characterized as the graphs that attain this bound.

The point-line geometry $(\mathcal{P},\mathcal{L})$ is called a finite projective plane of order $q$ if $|\mathcal{P}|=|\mathcal{L}|=q^{2}+q+1$, there exist $q+1$ points in each line, and there exist $q+1$ lines through each point. The incidence matrix of $(\mathcal{P},\mathcal{L})$ is the matrix indexed by $\mathcal{P}$ and $\mathcal{L}$ whose $(p,l)$ entry is 1 if $p\in l$, and 0 otherwise. An isomorphism $\varphi$ from $(\mathcal{P},\mathcal{L})$ to the dual plane $(\mathcal{L},\mathcal{P})$ is a polarity if $\varphi$ is an involution. We say $(\mathcal{P},\mathcal{L})$ admits polarity if there exists a polarity from $(\mathcal{P},\mathcal{L})$ to $(\mathcal{L},\mathcal{P})$. The classical finite projective planes admit a polarity. A finite projective plane $(\mathcal{P},\mathcal{L})$ admits a polarity if and only if the incidence matrix of $(\mathcal{P},\mathcal{L})$ can be symmetric. The symmetric incidence matrix of $(\mathcal{P},\mathcal{L})$ is the adjacency matrix of a $(q-1)$-regular multigraph with only 3 distinct eigenvalues which has loops. For $k\neq 2,3,7,57$, we show an upper bound $n\leq k^{2}-k+1$ for $k$-regular multigraphs of order $n$ with only 3 distinct eigenvalues. The equality holds if and only if the adjacency matrix of the graph is the symmetric incidence matrix of a finite projective plane of order $k-1$ that admits a polarity.

The paper is organized as follows. In Section 2, the linear programming bound [11] is generalized for connected regular multigraphs. We also give a certain improvement of the Moore bound with prescribed distinct eigenvalues. In Section 3, we prove the upper bound $n\leq k^{2}-k+1$ for $k\neq 2,3,7,57$. In Section 4, we show that the existence of a connected $k$-regular multigraph $G$ of order $k^{2}-k+1$ with only 3 distinct eigenvalues is equivalent to the existence of a finite projective plane $PG(2,k-1)$ that admits a polarity.

2 Bounds for regular multigraphs

Let $G$ be a multigraph $(V,E)$. For $v_{j}\in V$ and $e_{j}\in E$, a sequence $w_{p}=(v_{0},e_{1},v_{1},e_{2},v_{2},\ldots,v_{p-1},e_{p},v_{p})$ is a walk if $e_{j}=\{v_{j-1},v_{j}\}$ for each $j\in\{1,\ldots,p\}$. We shortly write a walk $w_{p}=(e_{1},\ldots,e_{p})$. The number $p$ is called the length of a walk. A walk $w_{p}$ is non-backtracking if there does not exist $j\in\{1,\ldots,p-1\}$ such that $e_{j}=e_{j+1}$, or $p=1$. A non-backtracking walk $w_{p}$ is a ${\it cycle}$ if $v_{0}=v_{p}$ and $v_{0},\ldots,v_{p-1}$ are distinct. The minimum length of cycles in $G$ is called the girth of $G$. If $G$ has a loop, then the girth of $G$ is 1. It is well known that the $(u,v)$-entry of $\boldsymbol{A}^{i}$ is the number of walks of length $i$ from $u$ to $v$. A multigraph $G$ is $k$-regular if $\sum_{v\in V}m(u,v)$ is $k$ for each $u\in V$.

Let $F_{i}^{(k)}$ denote a polynomial of degree $i$ defined by

[TABLE]

and

[TABLE]

for $i\geq 3$. Note that $F_{i}^{(k)}(k)=k(k-1)^{i-1}$ for $i\geq 1$.

Singleton [13] proved the following theorem only for $k$-regular simple graphs.

Theorem 2.1**.**

Let $G$ be a connected $k$-regular multigraph with adjacency matrix $\boldsymbol{A}$. Then the $(u,v)$-entry of $F_{i}^{(k)}(\boldsymbol{A})$ is the number of non-backtracking walks of length $i$ from $u$ to $v$.

Proof.

We use induction on $i$. Let $b_{uv}^{(i)}$ be the number of non-backtracking walks of length $i$ from $u$ to $v$. Let $f_{uv}^{(i)}$ be the $(u,v)$-entry of $F_{i}^{(k)}(\boldsymbol{A})$. For $i=1$, the assertion is trivial. For $i=2$, the $(u,v)$-entry $a^{(2)}_{uv}$ of $\boldsymbol{A}^{2}$ is the number of walks of length $2$ from $u$ to $v$. A walk that has backtracking must form $(e_{i},e_{i})$. The assertion follows from $b_{uv}^{(2)}=a^{(2)}_{uv}-k\delta_{uv}$, where $\delta$ is the Kronecker delta.

Suppose $f_{uv}^{(j)}=b_{uv}^{(j)}$ for each $j\in\{1,\ldots,i-1\}$. Since $F_{i}^{(k)}(\boldsymbol{A})=\boldsymbol{A}F_{i-1}^{(k)}(\boldsymbol{A})-(k-1)F_{i-2}^{(k)}(\boldsymbol{A})$, we have

[TABLE]

The value $\sum_{s\in V}b_{us}^{(1)}b_{sv}^{(i-1)}$ is the number of walks $(e_{1},\ldots,e_{p})$ such that $e_{1}=\{u,*\}$, $e_{p}=\{*,v\}$, and $(e_{2},\ldots,e_{p})$ is non-backtracking. We remove walks that have backtracking, namely the ones satisfying $e_{1}=e_{2}$. For given non-backtracking walk $(e_{3},\ldots,e_{p})$, the number of choices of $e_{1}$ is equal to $k-1$ because $e_{1}\neq e_{3}$. Therefore $f_{uv}^{(i)}=b_{uv}^{(i)}$ follows. ∎

Let $\boldsymbol{I}$ denote the identity matrix. Let $\boldsymbol{J}$ denote the matrix whose entries are all $1$. In [11] we proved the following theorem only for $k$-regular simple graphs.

Theorem 2.2**.**

Let $G$ be a connected $k$-regular multigraph of order $n$ with adjacency matrix $\boldsymbol{A}$. Let $\tau_{0},\ldots,\tau_{d}$ be the distinct eigenvalues of $\boldsymbol{A}$, where $\tau_{0}=k$. Let $f(x)$ be the polynomial defined by $f(x)=\sum_{i=0}^{s}f_{i}F_{i}^{(k)}(x)$ with a positive integer $s$ and real numbers $f_{0},\ldots,f_{s}$ such that $f_{0}>0$, $f_{i}\geq 0$ for each $i\in\{1,\ldots,s\}$. If $f(k)>0$ and $f(\tau_{j})\leq 0$ for each $j\in\{1,\ldots,d\}$, then

[TABLE]

Proof.

Since $\boldsymbol{A}$ is a real symmetric matrix, we have the spectral decomposition $\boldsymbol{A}=\sum_{i=0}^{d}\tau_{i}\boldsymbol{E}_{i}$, where $\boldsymbol{E}_{0}=(1/n)\boldsymbol{J}$. It follows that

[TABLE]

Taking the traces in (2.1), we have

[TABLE]

because $\boldsymbol{E}_{j}$ is positive semidefinite, and each entry in $F_{i}^{(k)}(\boldsymbol{A})$ is non-negative by Theorem 2.1. It therefore follows $n\leq f(k)/f_{0}$. ∎

Let $k_{i}=k(k-1)^{i-1}$ and $k_{0}=1$.

Theorem 2.3**.**

Let $G$ be a connected $k$-regular multigraph of order $n$ with adjacency matrix $\boldsymbol{A}$. Let $F(x)$ be the polynomial defined by

[TABLE]

for some real numbers $f_{0},\ldots,f_{s}$. If the entries of $F(\boldsymbol{A})$ are all positive, then

[TABLE]

Proof.

Since each $(u,v)$-entry of $F(\boldsymbol{A})$ is positive, there exists $i\in\{0,\ldots,d\}$ such that $f_{i}>0$ and the $(u,v)$-entry in $F_{i}^{(k)}(\boldsymbol{A})$ is positive. For each $u\in V$, the number of non-backtracking walks of length $i$ from $u$ is equal to $k_{i}$. Thus the number of non-zero entries in $F_{i}^{(k)}(\boldsymbol{A})$ is at most $nk_{i}$. Comparing the numbers of positive entries in the both sides in (2.2), it follows that

[TABLE]

This implies the theorem. ∎

Let $H_{G}(x)$ denote the Hoffman polynomial [7, 8] of a regular multigraph $G$, which is the polynomial of least degree satisfying $H_{G}(\boldsymbol{A})=\boldsymbol{J}$. If the distinct eigenvalues of $G$ are $\tau_{0}=k,\tau_{1},\ldots,\tau_{d}$ and the order of $G$ is $n$, then $H_{G}$ can be expressed by

[TABLE]

Corollary 2.4**.**

Let $G$ be a $k$-regular multigraph of order $n$, with only $d+1$ distinct eigenvalues $\tau_{0}=k,\tau_{1},\ldots,\tau_{d}$. Let $F_{G}(x)$ be the polynomial defined by $F_{G}(x)=\prod_{i=1}^{d}(x-\tau_{i})$. Then, from the expression $F_{G}(x)=\sum_{i=0}^{d}f_{i}F_{i}^{(k)}(x)$, it follows that $n\leq\sum_{i\in\{0,\ldots,d\}:f_{i}>0}k_{i}$.

Proof.

The polynomial $F_{G}(x)$ can be expressed by $F_{G}(x)=(\prod_{i=1}^{d}(k-\tau_{i})/n)H_{G}(x)$. Therefore, each entry of $F_{G}(\boldsymbol{A})=(\prod_{i=1}^{d}(k-\tau_{i})/n)\boldsymbol{J}$ is positive. Applying Theorem 2.3 to $F_{G}(x)$, we obtain the bound $n\leq\sum_{i\in\{0,\ldots,d\}:f_{i}>0}k_{i}$. ∎

If each $f_{i}$ is positive in Corollary 2.4, then the bound (2.3) coincides with the Moore bound.

3 Upper bound for regular multigraphs with three eigenvalues

In this section, we prove an upper bound for $k$-regular multigraphs with only $3$ distinct eigenvalues, which means Theorem 3.5. First we prove several lemmas to prove Theorem 3.5.

Lemma 3.1**.**

Let $G$ be a connected $k$-regular multigraph of order $n$ with only 3 distinct eigenvalues $k$, $\tau_{1}$, $\tau_{2}$. If $\tau_{1}+\tau_{2}\geq 0$, then $n\leq k^{2}-k+1$.

Proof.

The polynomial $F_{G}(x)=(x-\tau_{1})(x-\tau_{2})$ can be expressed by

[TABLE]

By $\tau_{1}+\tau_{2}\geq 0$ and Corollary 2.4, we have $n\leq k_{0}+k_{2}=k^{2}-k+1$. ∎

Lemma 3.2**.**

In a multigraph of maximum degree at most $k$, if a vertex $u$ is incident with a multiedge then there are at most $k^{2}-k$ vertices within distance two of $u$.

Proof.

Let $v$ be a vertex adjacent to $u$ with a multiedge. Then, it follows that

[TABLE]

where $\partial(u,w)$ is the distance between $u$ and $w$. ∎

Let $l_{v}$ denote the number of loops of $v\in V$.

Lemma 3.3**.**

Let $G$ be a connected $k$-regular multigraph of order $n$ with only 3 distinct eigenvalues. If $n>k^{2}-k+1$, then $G$ is simple and strongly regular.

Proof.

It suffices to show that $G$ is simple. Let $\tau_{1}$, $\tau_{2}$ be the distinct eigenvalues of $G$ with $\tau_{1},\tau_{2}\neq k$. By Lemma 3.1, we have $\tau_{1}+\tau_{2}<0$. By Lemma 3.2, $G$ has no multiedge. The Hoffman polynomial of $G$ can be expressed by

[TABLE]

It therefore follows that

[TABLE]

where $\boldsymbol{A}$ is the adjacency matrix of $G$. Comparing the $(v,v)$-entry of the both sides in (3.1), we obtain

[TABLE]

The value $l_{v}^{2}-(\tau_{1}+\tau_{2}+1)l_{v}$ is constant for each $v\in V$. If $l_{v}>0$ for each $v\in V$, then

[TABLE]

which contradicts our assumption. We may suppose some $v\in V$ satisfies $l_{v}=0$. This implies that $l_{u}^{2}-(\tau_{1}+\tau_{2}+1)l_{u}=0$, namely $l_{u}=0$ or $l_{u}=\tau_{1}+\tau_{2}+1$ for each $u\in V$. Since $\tau_{1}+\tau_{2}<0$ holds, it follows that $l_{u}=\tau_{1}+\tau_{2}+1<1$ and $l_{u}=0$ for each $u\in V$. ∎

Lemma 3.4**.**

Let $G$ be a connected $k$-regular multigraph of order $n$ with only 3 distinct eigenvalues. If $n>k^{2}-k+1$ and $k\geq 3$, then there does not exist $G$ except for Moore graphs. If $n>k^{2}-k+1$ and $k=2$, then $G$ is the cycle graph of order 4 or 5.

Proof.

By Lemma 3.3, $G$ is strongly regular, and let $(n,k,\lambda,\mu)$ be the parameters of $G$. The assertion clearly holds for $k=2$. Suppose $k\geq 3$. Let $\tau_{1}$, $\tau_{2}$ be the distinct eigenvalues of $G$ with $\tau_{1},\tau_{2}\neq k$. For connected strongly regular graphs, it follows that $\mu\neq 0$. If $\mu\geq 2$, then

[TABLE]

from $k\geq 3$. Thus $\mu=1$. If $\lambda=0$, then $G$ is a Moore graph. If $\lambda=1$, then $G$ gives rise to a projective plane with a polarity containing no absolute points, which is not possible [6]. If $\lambda>1$, then there exists an integer $s$ such that $k=s(\lambda+1)$ and $n=1+s(\lambda+1)+s(s-1)(\lambda+1)^{2}$ [6], which gives

[TABLE]

Theorem 3.5**.**

Let $G$ be a connected $k$-regular multigraph of order $n$ with only 3 distinct eigenvalues. Then, one has $n\leq k^{2}-k+1$ for $k\neq 2,3,7,57$.

Proof.

By Lemma 3.4, if $n>k^{2}-k+1$, then $G$ is a Moore graph. There does not exist a Moore graph except for $k\in\{2,3,7,57\}$ [2, 5]. This implies the theorem. ∎

4 Largest regular multigraphs with three eigenvalues

For $k\neq 2,3,7,57$, we have $n\leq k^{2}-k+1$ by Theorem 3.5. The largest multigraphs are constructed from finite projective planes. Refer to [12] for projective planes. Suppose $q=k-1$ is a prime power. Let $\mathbb{F}_{q}$ be the finite field of order $q$. Let $V_{q}$ be a 3-dimensional vector space over $\mathbb{F}_{q}$. Let $\mathcal{P}_{q}$ (resp. $\mathcal{L}_{q}$) be the set of all 1-dimensional (resp. 2-dimensional) subspaces of $V_{q}$. Note that $|\mathcal{P}_{q}|=|\mathcal{L}_{q}|=q^{2}+q+1=k^{2}-k+1$. A point $p\in\mathcal{P}_{q}$ is incident with a line $l\in\mathcal{L}_{q}$ if $p\subset l$. The point-line geometry $(\mathcal{P}_{q},\mathcal{L}_{q})$ is called a classical finite projective plane. Let $\Gamma_{q}$ denote the incidence graph of $(\mathcal{P}_{q},\mathcal{L}_{q})$. The graph $\Gamma_{q}$ is bipartite and its adjacency matrix can be expressed by

[TABLE]

where $\boldsymbol{A}$ is the incidence matrix of $(\mathcal{P}_{q},\mathcal{L}_{q})$. The set of eigenvalues of $\Gamma_{q}$ is $\{\pm(q+1),\pm\sqrt{q}\}$. We may suppose $\boldsymbol{A}$ is symmetric by the correspondence $\{(p,l)\in\mathcal{P}_{q}\times\mathcal{L}_{q}\colon\,p\perp l\}$, where we use the usual inner product of $V_{q}$. This implies that $\boldsymbol{A}$ is the adjacency matrix of a $(q+1)$-regular graph $G_{q}$ and has only 3 distinct eigenvalues $\{q+1,\pm\sqrt{q}\}$. Note that $G_{q}$ has loops. For any prime power $q$, the graph $G_{q}$ is a largest $k$-regular multigraph attaining the bound from Theorem 3.5.

The following is a necessary condition for a graph to attain the bound from Theorem 3.5.

Lemma 4.1**.**

Let $G$ be a connected $k$-regular multigraph of order $n$ with only 3 distinct eigenvalues $k$, $\tau_{1}$, $\tau_{2}$. If $n=k^{2}-k+1$, then $G$ has a loop and no multiedge, $l_{v}\in\{0,1\}$ for each $v\in V$, and $\tau_{1}+\tau_{2}=0$.

Proof.

By $n=k^{2}-k+1$ and Lemma 3.2, there does not exist a multiedge in $G$. If there exists $v\in V$ such that $l_{v}>1$, then

[TABLE]

Thus $l_{v}\leq 1$ for each $v\in V$. As we see in the proof of Lemma 3.3, there exists $v\in V$ such that $l_{v}=0$. Moreover $l_{u}^{2}-(\tau_{1}+\tau_{2}+1)l_{u}=0$, namely $l_{u}=0$ or $l_{u}=\tau_{1}+\tau_{2}+1$ for each $u\in V$. If there exists $u\in V$ such that $l_{u}=\tau_{1}+\tau_{2}+1=1$, then $\tau_{1}+\tau_{2}=0$. Assume $l_{u}=0$ for each $u\in V$. Now $G$ is a strongly regular graph with parameters $(v,k,\lambda,\mu)$. If $\mu\geq 2$, then (3.2) holds. The last equality in (3.2) is attained only for $(n,k)=(7,3)$, which is impossible. Thus $\mu=1$. By the same argument as the last part in the proof of Lemma 3.4, for any $\lambda$ there does not exist $G$ of order $k^{2}-k+1$. ∎

The following is the main theorem in this section.

Theorem 4.2**.**

The existence of a connected $k$-regular multigraph $G$ of order $k^{2}-k+1$ with only 3 distinct eigenvalues is equivalent to the existence of a finite projective plane $PG(2,k-1)$ that admits a polarity.

Proof.

If a finite projective plane $PG(2,k-1)$ that admits a polarity exists, then the incidence matrix can be symmetric, and it is the adjacency matrix of a $k$-regular multigraph of order $k^{2}-k+1$ with only 3 distinct eigenvalues.

Let $G$ be a connected $k$-regular multigraph of order $k^{2}-k+1$ with only 3 distinct eigenvalues. By Lemma 4.1, the eigenvalues are $k,\pm\tau$, and the bipartite double graph $G^{\prime}$ of $G$ is simple. Since the eigenvalues of $G^{\prime}$ are $\pm k,\pm\tau$, the diameter of $G^{\prime}$ is at most $3$. Thus the graph $G^{\prime}$ attains the bipartite Moore bound $n\leq 2(1+(k-1)+(k-1)^{2})=2(k^{2}-k+1),$ and the girth of $G^{\prime}$ is $6$. The graph $G^{\prime}$ is the cage $v(k,6)$, and $G^{\prime}$ must be the incidence graph of a finite projective plane $PG(2,k-1)$ (see [3, Section 6.9]). Now the incidence matrix of the projective plane $PG(2,k-1)$ is symmetric, and hence there exists a polarity on it. ∎

By Theorem 4.2, largest $k$-regular multigraphs with only 3 distinct eigenvalues are obtained for a prime power $q=k-1$. Open cases of small degrees are $k=11,13,15,16,19,21,22,23,\ldots$. For $q\equiv 1,2\pmod{4}$, if a projective plane of order $q$ exists, then $q$ is the sum of two integral squares [4]. Therefore for $k=13$ a projective plane of order $14$ does not exist. For $k=11$, there does not exist a finite projective plane of order $10$ by a computer search [10]. If $\boldsymbol{A}$ is the adjacency matrix of some $k$-regular multigraph, then $\boldsymbol{A}+t\boldsymbol{I}$ is that of a $(k+t)$-regular multigraph, and has the same number of distinct eigenvalues as $\boldsymbol{A}$. This construction gives a candidate of the largest graph when a projective plane does not exist.

Acknowledgments. The author is supported by JSPS KAKENHI Grant Numbers 16K17569, 26400003, 17K05155, 18K03396, and 19K03445. The author would like to thank the four anonymous referees for their valuable suggestions which helped to improve the earlier version of this paper.