Non-existence of antipodal cages of even girth

TL;DR

This paper proves that antipodal $(k,g)$-cages with even girth $g \,\geq\,14$ and small excess do not exist, using spectral analysis of their adjacency and distance matrices.

Contribution

It establishes the non-existence of certain antipodal cages with even girth and small excess through spectral analysis methods.

Findings

Proves non-existence of antipodal $(k,g)$-cages for $g\geq14$ and small excess.

Provides spectral relations between adjacency and distance matrices.

Extends previous results on bipartite cages and antipodal properties.

Abstract

The Moore bound $M(k,g)$ is a lower bound on the order of $k$-regular graphs of girth $g$ (denoted $(k,g)$-graphs). The excess $e$ of a $(k,g)$-graph of order $n$ is the difference $n-M(k,g).$ A $(k,g)$-cage is a $(k,g)$-graph with the fewest possible number of vertices, among all $(k,g)$-graphs. A graph of diameter $d$ is said to be antipodal if, for any vertices $u, v, w$ such that $d(u,v)=d$ and $d(u, w)=d$, it follows that $d(v, w)=d$ or $v=w.$ In [4] Biggs and Ito proved that any $(k,g)$-cage of even girth $g=2d\geq6$ and excess $e\leq k-2$ is a bipartite graph of diameter $d+1.$ In this paper we treat the $(k,g)$-cages of even girth and excess $e\leq k-2.$ Based on a spectral analysis we give a relation between the eigenvalues of the adjacency matrix $A$ and the distance matrix $A_{d+1}$ of such cages. Moreover, following the methodology used in [4] and [13], we prove the non-existence of the antipodal $(k,g)$-cages of excess $e$, where $k\geq e+2\geq4$ and $g=2d\geq14.$

Figures

Equations

$$M(k,g)=\left\{\begin{array}[]{lc}1+k+k(k-1)+\cdot\cdot\cdot+k(k-1)^{(g-3)/2},&g\mbox{ odd, }\\ 2\left(1+(k-1)+\cdot\cdot\cdot+(k-1)^{(g-2)/2}\right),&g\mbox{ even. }\end{array}\right.$$

$$P_{i+1}(x)=xP_{i}(x)-(k-1)P_{i-1}(x)\mbox{ for }\left\{\begin{array}[]{lc}i\geq 1,&\mbox{ if }P_{i}=G_{i},\\ i\geq 2,&\mbox{ if }P_{i}=F_{i},\\ i\geq 1,&\mbox{ if }P_{i}=H_{i}.\end{array}\right.$$

$$F_d(A)=k{A}_d-A{A}_{d+1}.$$

$$kJ=(A+kI)(H_{d-1}(A)+A_{d+1}).$$

$$H_{d-1}(\theta )=-\lambda ,$$

$$H_{d-1}(\theta )-ϵ=0,$$

$$H_{d-1}(x)=(-s{)}^{d-1}\frac{\sin d\varphi }{\sin \varphi }.$$

$$\sin \alpha -{\eta }_i{s}^{-d+1}\sin \left(\frac{i\pi -\alpha }{d}\right)=0,$$

$$B_{D}=\left(\begin{array}[]{cccccccc}0&1&&&&&&\\ k&0&1&&&&0&\\ &k-1&0&1&&&&\\ &&k-1&0&1&&&\\ &&&&\ddots&\ddots&&\\ &&&&\ddots&&&\\ &0&&&&k-1&0&k\\ &&&&&&k-1&0\\ \end{array}\right)$$

$$m(\theta )=\frac{nek(k-1)H_{d-2}(\theta )}{2ϵ(2ϵ+\frac{e}{2}-1)H_{d-1}^{{}^{\prime }}(\theta )(k^2-{\theta }^2)}.$$

$$tr({\xi }_{\theta }(A))=tr(A^{2d-1})+a_{2d-2}tr(A^{2d-2})+...+a_{1}tr(A)+a_{0}tr(I_n).$$

$$tr({\xi }_{\theta }(A))=n({\xi }_{\theta }(B_d){)}_{0,0}.$$

$${\xi }_{\theta }(B_d)=-\frac{e}{2}\frac{B_d^2-k^2{I}_n}{B_d-\theta I_n}.$$

$$L_d(B_d)=-H_{d-1}(\theta )\frac{B_d^2-k^2{I}_n}{B_d-\theta I_n}=-ϵ\frac{B_d^2-k^2{I}_n}{B_d-\theta I_n}.$$

$$m(\theta )=\frac{ne}{2ϵ(2ϵ+\frac{e}{2}-1)}\frac{(L_d(B_d){)}_{0,0}}{H_{d-1}^{{}^{\prime }}(\theta )({\theta }^2-k^2)}.$$

$$m(\theta )=\frac{nek(k-1)}{2ϵ(2ϵ+\frac{e}{2}-1)}\frac{H_{d-2}(\theta )}{H_{d-1}^{{}^{\prime }}(\theta )(k^2-{\theta }^2)}.$$

$$f(z)=\frac{4s^2(1-z^2)}{k^2-4s^2{z}^2};$$

$$g_1(z)=\frac{k(k-1)(\sqrt{1-s^{-2d+2}(1-z^2)}+s^{-d+1}z)}{d\sqrt{1-s^{-2d+2}(1-z^2)}+s^{-d+1}z};$$

$$g_2(z)=\frac{k(k-1)(\sqrt{1-\frac{e^2}{4}{s}^{-2d+2}(1-z^2)}-\frac{e}{2}{s}^{-d+1}z)}{d\sqrt{1-\frac{e^2}{4}{s}^{-2d+2}(1-z^2)}-\frac{e}{2}{s}^{-d+1}z};$$

$$g_3(z)=\frac{k(k-1)(\sqrt{1-\frac{e^2}{4}{s}^{-2d+2}(1-z^2)}+\frac{e}{2}{s}^{-d+1}z)}{d\sqrt{1-\frac{e^2}{4}{s}^{-2d+2}(1-z^2)}+\frac{e}{2}{s}^{-d+1}z}.$$

$$m(\theta_{i})=\frac{ne}{4s^{2}(\frac{e}{2}+1)}f(\cos\phi_{i})g_{1}(\eta_{i}\cos\phi_{i}),\mbox{ if $\epsilon=1$};$$

$$m(\theta_{i})=\frac{n}{2s^{2}(\frac{e}{2}+1)}f(\cos\phi_{i})g_{2}(\cos\phi_{i}),\mbox{ if $\epsilon=-\frac{e}{2}$ and $i$ is odd};$$

$$m(\theta_{i})=\frac{n}{2s^{2}(\frac{e}{2}+1)}f(\cos\phi_{i})g_{3}(\cos\phi_{i}),\mbox{ if $\epsilon=-\frac{e}{2}$ and $i$ is even}.$$

$$H_{d-1}^{{}^{\prime }}({\theta }_i)=\frac{(-s{)}^{d-1}(-1{)}^i}{2s{\sin }^2{\varphi }_i}(d\cos {\alpha }_i+{\eta }_i{s}^{-d+1}\cos {\varphi }_i).$$

$$m({\theta }_i)=\frac{nek(k-1)\sin {\varphi }_i\sin ({\varphi }_i+{\alpha }_i)}{ϵ(2ϵ+\frac{e}{2}-1)(k^2-{\theta }_i^2)(d\cos {\alpha }_i+{\eta }_i{s}^{-d+1}\cos {\varphi }_i)}.$$

$$m({\theta }_i)=\frac{ne{\sin }^2{\varphi }_i}{ϵ(2ϵ+\frac{e}{2}-1)(k^2-{\theta }_i^2)}\frac{k(k-1)(\cos {\alpha }_i+{\eta }_i{s}^{-d+1}\cos {\varphi }_i)}{(d\cos {\alpha }_i+{\eta }_i{s}^{-d+1}\cos {\varphi }_i)}.$$

$$m({\theta }_i)=\frac{ne}{4s^2ϵ(2ϵ+\frac{e}{2}-1)}\frac{4s^2(1-{\cos }^2{\varphi }_i)}{k^2-4s^2{\cos }^2{\varphi }_i}\frac{k(k-1)(\sqrt{1-{\eta }_i^2{s}^{-2d+2}(1-{\cos }^2{\varphi }_i)}+{\eta }_i{s}^{-d+1}\cos {\varphi }_i)}{(d\sqrt{1-{\eta }_i^2{s}^{-2d+2}(1-{\cos }^2{\varphi }_i)}+{\eta }_i{s}^{-d+1}\cos {\varphi }_i)}.$$

$$g_1^{{}^{\prime }}(z)=\frac{k(k-1)(d-1)s^{-d+1}(1-s^{-2d+2})}{\sqrt{1+s^{-2d+2}(-1+z^2)}(d\sqrt{1+s^{-2d+2}(-1+z^2)}+s^{-d+1}z{)}^2}>0.$$

$$g_2^{{}^{\prime }}(z)=\frac{-\frac{e}{2}{s}^{-d+1}k(k-1)(d-1)(1-\frac{e^2}{4}{s}^{-2d+2})}{\sqrt{1+\frac{e^2}{4}{s}^{-2d+2}(-1+z^2)}(d\sqrt{1+\frac{e^2}{4}{s}^{-2d+2}(-1+z^2)}-\frac{e}{2}{s}^{-d+1}z{)}^2}.$$

$$f(\cos\phi_{2})<f(\cos\phi_{i})\mbox{ for $3\leq i\leq d-3$.}$$

Full text

Keywords: $k$-regular graphs, antipodal cages, excess, multiplicities

1 Introduction

A $(k,g)$-graph is a $k$-regular graph having girth $g$. A $(k,g)$-cage is a $(k,g)$-graph of smallest order. The Cage Problem or Degree/Girth Problem calls for finding cages; Tutte was the first to study $(k,g)$-cages, [16]. A $(k,g)$-graph exists for any pair $(k,g),$ where $k\geq 2$ and $g\geq 3$, see [8] and [14]. It is well known that the $(k,g)$-graphs have at least $M(k,g)$ vertices, where

[TABLE]

If $G$ is a $(k,g)$-graph of order $n$, then we define the excess $e$ of $G$ to be $n-M(k,g).$ The graphs whose orders are equal to $M(k,g)$ (excess [math]) are called Moore graphs. Their classification has been completed except for the case $k=57$ and $g=5.$ The Moore graphs exist if $k=2$ and $g\geq 3$, $g=3$ and $k\geq 2$, $g=4$ and $k\geq 2$, $g=5$ and $k=2,3,7$, or $g=6,8,12$ and a generalized $n$-gon of order $k-1$ exists, see [1], [7] and [9].

The following three results concern the graphs of even girth.

Theorem 1.1** (Biggs and Ito [4])**

Let $G$ be a $(k,g)$-cage of girth $g=2d\geq 6$ and excess $e$. If $e\leq k-2$, then $e$ is even and $G$ is bipartite of diameter $d+1$.

It is known that these graphs are partially distance-regular. More about almost-distance-regular graphs, see [5]. For the next theorem, let $D(k,2)$ denote the incidence graph of a symmetric $(v,k,2)$-design.

Theorem 1.2** (Biggs and Ito [4])**

Let $G$ be a $(k,g)$-cage of girth $g=2d\geq 6$ and excess $2$. Then $g=6$, $G$ is a double-cover of $D(k,2)$, and $k\not\equiv 5,7\!\!\pmod{8}$.

Theorem 1.3** (Jajcayová, Filipovski and Jajcay [12])**

Let $k\geq 6$ and $g=2d>6$. No $(k,g)$-graphs of excess $4$ exist for parameters $k,g$ satisfying at least one of the following conditions:

$g=2p$, with $p\geq 5$ a prime number, and $k\not\equiv 0,1,2\!\!\pmod{p}$;

• 2)

$g=4\cdot 3^{s}$* such that $s\geq 4$, and $k$ is divisible by $9$ but not by $3^{s-1}$;*

• 3)

$g=2p^{2}$, with $p\geq 5$ a prime number, and $k\not\equiv 0,1,2\!\!\pmod{p}$ and $k$ even;

• 4)

$g=4p$, with $p\geq 5$ a prime number, and $k\not\equiv 0,1,2,3,p-2\!\!\pmod{p}$;

• 5)

$g\equiv 0\!\!\pmod{16}$, and $k\equiv 3\!\!\pmod{g}.$

Let $k\geq 4,g=2d\geq 6$ and let $G$ be a $(k,g)$-cage of excess $e\leq k-2$ and order $n$. Due to Theorem 1.1, we conclude that $G$ is a bipartite graph of diameter $d+1.$ Let $A$ be its adjacency matrix. For the integers $i$ with $0\leq i\leq d+1$, the $i$-distance matrix $A_{i}$ of $G$ is an $n\times n$ matrix such that the entry in position $(u,v)$ is $1$ if the distance between the vertices $u$ and $v$ is $i$, and zero otherwise. Using the spectral considerations as in [10], in Section 2 we prove that the eigenvalues of $A(A_{1})$, other than $\pm k$, are the roots of the polynomial $H_{d-1}(x)+\lambda$; Theorem 2.3. Here, $H_{d-1}$ is the Dickson polynomial of the second kind with parameter $k-1$ and degree $d-1$, and $\lambda$ is an eigenvalue of the distance matrix $A_{d+1}$.

A graph of diameter $d$ is said to be antipodal if, for any vertices $u,v,w$ such that $d(u,v)=d$ and $d(u,w)=d$, it follows that $d(v,w)=d$ or $v=w,$ (see [3]).

Among the trivially antipodal graphs let us mention the $n$-dimensional cubes $Q_{n}$. These graphs are bipartite and have the antipodal property, since every vertex of $Q_{n}$ has a unique vertex at maximum distance from it. Also, for $n\geq 2$, the complete bipartite graph $K_{n,n}$ is antipodal. Here the antipodal partition is the same as the bipartition. The dodecahedron is an example of trivially antipodal, but not bipartite graph. Examples of graphs which are non-trivially antipodal and not bipartite are the complete tripartite graphs $K_{n,n,n}$, which have diameter $2$, and the line graph of Petersen’s graph, which has diameter $3$. Motivated by Theorem 1.4, in this paper we address the question of the existence of the antipodal $(k,g)$-cages of even girth and excess $e\leq k-2.$ Employing the methodology used in [2], [4] and [13], we prove the non-existence of the antipodal $(k,g)$-cages of excess $e$, with $k\geq e+2\geq 4$ and $g=2d\geq 14;$ Theorem 4.2.

Theorem 1.4** (Bannai and Ito [2])**

For $d\geq 3,$ there exist no antipodal regular graphs with diameter $d+1$ and girth $2d+1.$

2 On $(k,g)$-cages of even girth and excess $e\leq k-2$

Let $k,g,d$ and $e$ be positive integers such that $k\geq e+2$ and $g=2d\geq 6$. Let $G$ be a $(k,g)$-cage of excess $e$; Theorem 1.1 asserts that $e$ is even and $G$ is a bipartite graph of diameter $d+1.$ Let $f=\{u,v\}$ be an arbitrary edge of $G$. Let ${\mathcal{T}}_{u}$ be the subgraph of $G$ induced by the set of vertices $x\in V(G)$ such that $d(u,x)\leq\frac{g-2}{2}$ and $d(v,x)=d(u,x)+1$. It is easy to see that ${\mathcal{T}}_{u}$ is a tree of depth $\frac{g-2}{2}$ rooted at $u$. In the same way we can define the tree ${\mathcal{T}}_{v}$ to be the subgraph of $G$ induced by the set of vertices $x\in V(G)$ such that $d(v,x)\leq\frac{g-2}{2}$ and $d(u,x)=d(v,x)+1$. Since $G$ is of girth $g=2d$, the trees ${\mathcal{T}}_{u}$ and ${\mathcal{T}}_{v}$ are disjoint. Let ${\mathcal{T}}_{uv}$ be the union of the trees ${\mathcal{T}}_{u}$ and ${\mathcal{T}}_{v}$ and the edge $f$. We call the graph ${\mathcal{T}}_{uv}$ a Moore tree of $G$ rooted at $f$. The graph $G$ must contain $e$ additional vertices $w_{1},w_{2},...,w_{e-1},w_{e}$, which do not belong to ${\mathcal{T}}_{uv}$, that is, $d(w_{i},u)>\frac{g-2}{2}$ and $d(w_{i},v)>\frac{g-2}{2}$, for each $1\leq i\leq e.$ We call these vertices the excess vertices with respect to $f$ and denote this set $X_{f}=\{w_{1},w_{2},...,w_{e-1},w_{e}\}$; we call the edges not contained in the Moore tree of $G$ horizontal edges. Since $G$ is a bipartite graph, it contains no odd cycle; consequently there exists no edge between the excess vertices of the same partite set. Moreover, in order to balance the Moore tree ${\mathcal{T}}_{uv}$ and paring out the horizontal edges of $G$, we easily observe that half ($\frac{e}{2}$) of the excess vertices belong to the first, and the other half to the second partite set of $G.$ It implies that for each vertex of $V(G)$ there exist exactly $\frac{e}{2}$ vertices at distance $d+1$ from it.

In order to study the spectral properties of $G$, we define the following polynomials:

$G_{0}(x)=1,\;G_{1}(x)=x+1;$

$F_{0}(x)=1,\;F_{1}(x)=x,\;F_{2}(x)=x^{2}-k;$

$H_{-2}(x)=-\frac{1}{k-1},\;H_{-1}(x)=0,\;H_{0}(x)=1,\;H_{1}(x)=x;$

[TABLE]

The above defined polynomials have a close connection to the properties of a graph $G$. Namely, for $l<g$, the element $(F_{l}(A))_{x,y}$ counts the number of paths of length $l$ joining vertices $x$ and $y$ of $G$. Moreover, $G_{l}(A)$ counts the number of paths of length at most $l$ joining pairs of vertices in $G$. For more information about these polynomials see [15].

The next lemma is a generalization on Lemma 5 from [10], where it is considered cages of even girth and excess $4$.

Lemma 2.1

Let $k\geq e+2$ and $g=2d\geq 6,$ and let $G$ be a $(k,g)$-cage of excess $e$. If $A$ is the adjacency matrix of $G$, then

[TABLE]

Proof. Let $f=\{u,v\}$ be a base edge of the Moore tree and let $f_{1}=\{w_{1},w_{2}\},f_{2}=\{w_{3},w_{4}\},...,f_{\frac{e}{2}}=\{w_{e-1},w_{e}\}$ be the edges of the subgraph induced by $X_{f}$. Also, let us assume that $d(u,w_{1})=d(u,w_{3})=...=d(u,w_{e-1})=d$ and $d(u,w_{2})=d(u,w_{4})=...=d(u,w_{e})=d+1.$ Let $l_{i}$ be the number of edges between $w_{i}$ and the leaves of ${\mathcal{T}}_{u}$ and ${\mathcal{T}}_{v}$, where $1\leq i\leq e.$ We consider the case when the excess vertices do not share common neighbour among the leaves of ${\mathcal{T}}_{u}$ and ${\mathcal{T}}_{v}$. The opposite case one can prove in a similar way. By the definition of $F_{i}(x)$, we have $(F_{d}(A))_{u,w_{i}}=l_{i}$, for each odd $i$, $1\leq i\leq e-1.$ Considering the vertices at distance $d$ from $u$, there are also the $(k-1)^{d-1}$ leaves of ${\mathcal{T}}_{v}$. For $l_{2}+l_{4}+...+l_{e}$ of these vertices, there exist $k-1$ paths of length $d$ from $u$ to them. Namely, they are the vertices adjacent to $w_{2},w_{4},...,w_{e-2}$ or $w_{e}.$ For all the other leaves, there are $k$ paths between them and $u$. Thus, $(F_{d}(A))_{u,s}=0$ if $d(u,s)\neq d$, $(F_{d}(A))_{u,s}=k$ if $s$ is a leaf of ${\mathcal{T}}_{v}$ and not adjacent to $w_{2},w_{4},...,w_{e}$, $(F_{d}(A))_{u,s}=k-1$ if $s$ is a leaf of ${\mathcal{T}}_{v}$ and adjacent to one of $w_{2},w_{4},...,w_{e}$, and $(F_{d}(A))_{u,w_{i}}=l_{i}$, for each odd $i$; $1\leq i\leq e-1.$ For the matrix $kA_{d}$ we have $(kA_{d})_{u,s}=k$ if $d(u,s)=d$ and $(kA_{d})_{u,s}=0$ if $d(u,s)\neq d.$ Now, let $s$ be a vertex of $G$ such that $d(u,s)=d$ and $s$ is adjacent to one of $w_{2},w_{4},...,w_{e}.$ If $s$ is a vertex among the vertices $w_{1},w_{3},...,w_{e-1}$, then it is easy to see that $(AA_{d+1})_{u,s}=k-l_{i}.$ On the other hand, since $s$ is adjacent to ${\mathcal{T}}_{u}$ through $k-2$ different horizontal edges, it follows that, between the $k-1$ branches of ${\mathcal{T}}_{u}$, there exists one sub-branch that is not adjacent to $s$ through a horizontal edge. Let $s_{1}$ be the root of that sub-branch. Then, $d(s,s_{1})=d+1$ and $d(u,s_{1})=1$, which implies $(A)_{u,s_{1}}=1$ and $(A_{d+1})_{s_{1},s}=1.$ Let $s_{i}$, $2\leq i\leq\frac{e}{2}$ be the remaining vertices at distance $d+1$ from $s$. Because all neighbours of $u$, except $s_{1}$, are at distance smaller than $d+1$ from $s$, we have $(A)_{u,s_{i}}=0$ and $(A_{d+1})_{s_{i},s}=1,$ for $2\leq i\leq\frac{e}{2}.$ Thus $(AA_{d+1})_{u,s}=1.$ If $s$ is a vertex of $G$ such that $d(u,s)=d$ and $s$ is not adjacent to $w_{2},w_{4},...,w_{e}$, then the distance between $s$ and the neighbours of $u$ is $d-1$. In this case, $(AA_{d+1})_{u,s}=0.$ If $d(u,s)\neq d$, then the distance between $s$ and the neighbours of $u$ is different from $d+1$, and therefore $(AA_{d+1})_{u,s}=0.$ The required identity follows from summing up the above conclusions.

Based on the previous lemma and the properties of the polynomials $G_{i}$, $H_{i}$ and $F_{i}$, we obtain the next two results. Theorem 2.3 is the main result in this section; it gives a relationship between the eigenvalues of the matrices $A$ and $A_{d+1}$. We omitted their proofs because they follow analoglously like in Lemma 6 and Theorem 7 from [10].

Lemma 2.2

Let $k\geq e+2\geq 4$ and $g=2d\geq 6$, and let $G$ be a $(k,g)$-cage of excess $e$. If $A$ is the adjacency matrix of $G$ and $J$ is the all-ones matrix, then

[TABLE]

Theorem 2.3

If $\theta(\neq\pm k)$ is an eigenvalue of $A$, then

[TABLE]

where $\lambda$ is an eigenvalue of $A_{d+1}$.

3 Spectral analysis of the antipodal cages of even girth and small excess

In this section we study the spectral properties of the antipodal $(k,g)$-cages of even girth $g=2d\geq 6$ and excess at most $k-2$. Let $G$ be such graph, $A$ be its adjacency matrix and let $n$ be the order of $G$. Recall, $G$ is a bipartite graph of diameter $d+1$. Let $V_{1}$ and $V_{2}$ be the partitions of $G$. If $d$ is an even number, then any two vertices of $V(G)$ at distance $d+1$ belong to a different partite set. Clearly, this case is not possible considering antipodal bipartite graphs. Therefore, for the rest of the paper we assume $d$ odd. Since for each vertex $u\in V(G)$ there exist exactly $\frac{e}{2}$ vertices at diameter distance $d+1$, (they are the excess vertices of the same partite set), we observe that the antipodal graph of $G$ is a disjoint union of $K_{\frac{e}{2}+1}$-complete graphs, and consequently, the distance matrix $A_{d+1}$ is an adjacency matrix of a disjoint union of $K_{\frac{e}{2}+1}$-complete graphs. Let $c$ be the number of such complete graphs. Obviously $c=\frac{2n}{e+2}.$ The spectrum of the disjoint union of $c$ complete graphs of order $\frac{e}{2}+1$ is known and determined by $\{(\frac{e}{2})^{c},(-1)^{n-c}\}$ (see Propos. 6 in [6]). Applying this result in Theorem 2.3, we are in a position to determine the spectrum of $A$.

Theorem 3.1

If $\theta(\neq\pm k)$ is an eigenvalue of $A$, then

[TABLE]

where $\epsilon=-\frac{e}{2},1$.

The roots of $H_{d-1}(x)$ are equal to $2\sqrt{k-1}\cos\frac{i\pi}{d}$ for $i=1,...,d-1$, (see [15]). Therefore we assume $x=-2\sqrt{k-1}\cos\phi,0<\phi<\pi$. Let $s=\sqrt{k-1}.$ Then we have

[TABLE]

Putting $\phi=\frac{i\pi-\alpha}{d}$, as suggested in [2] and [4], we transform the equation (3) as follows

[TABLE]

where $\eta_{i}=\epsilon(-1)^{d+i}.$ The following result follows similarly as Lemma 3.3 from [4] and Lemma 2.2 from [13].

Lemma 3.2

The equation (3) has $d-1$ distinct roots $\theta_{1}<\theta_{2}<...<\theta_{d-1}$, with $\theta_{i}=-2s\cos\phi_{i},(0<\phi_{i}<\pi).$ If we set $\phi_{i}=\frac{i\pi-\alpha_{i}}{d}$ then

$0<\alpha_{i}<min\{s^{-d+1}\phi_{i},s^{-d+1}(\pi-\phi_{i})\}$* if $\eta_{i}=1;$*

$max\{-s^{-d+1}\phi_{i},-s^{-d+1}(\pi-\phi_{i})\}<\alpha_{i}<0$* if $\eta_{i}=-1;$*

$0<\alpha_{i}<min\{\frac{e}{2}s^{-d+1}\phi_{i},\frac{e}{2}s^{-d+1}(\pi-\phi_{i})\}$* if $\eta_{i}=\frac{e}{2};$*

$max\{-\frac{e}{2}s^{-d+1}\phi_{i},-\frac{e}{2}s^{-d+1}(\pi-\phi_{i})\}<\alpha_{i}<0,$* if $\eta_{i}=-\frac{e}{2}.$*

From the bounds of $\alpha_{i}$ we derive the bounds of $\phi{{}_{i}}$ as follows.

$\frac{i\pi}{d+s^{-d+1}}<\phi_{i}<\frac{i\pi}{d}$* if $\eta_{i}=1$;*

$\frac{i\pi}{d}<\phi_{i}<\frac{i\pi}{d-s^{-d+1}}$* if $\eta_{i}=-1$;*

$\frac{i\pi}{d+\frac{e}{2}s^{-d+1}}<\phi_{i}<\frac{i\pi}{d}$* if $\eta_{i}=\frac{e}{2};$*

$\frac{i\pi}{d}<\phi_{i}<\frac{i\pi}{d-\frac{e}{2}s^{-d+1}}$* if $\eta_{i}=-\frac{e}{2}.$*

We claim that $tr(A^{q})=n(B_{d}^{q})_{0,0}$ for $q=0,1,...,2d-1,$ where

[TABLE]

is the $(D+1)\times(D+1)$ intersection matrix of a Moore bipartite graph of degree $k$, diameter $D$ and of girth $2D$, (see [13]). If $q<g(G)$, the number of closed walks of length $q$ that start from a fixed vertex $u$ is independent of the vertex $u$ and the excess. Furthermore, the entry $(B_{\lceil\frac{g(G)}{2}\rceil}^{q})_{0,0}$ gives this number, where $(B_{i}^{q})_{0,0}$ is the $(0,0)$-entry of $B_{i}^{q},$ (see [11]). The number of closed walks of length $q$ in $G$ is given by $tr(A^{q})$. Since $G$ is a bipartite graph, it follows that $G$ contains no closed walk of odd length. Thus, $tr(A^{q})=n(B_{d}^{q})_{(0,0)}$ for $q=1,3,...,2d-3,2d-1.$ Moreover, since the girth of $G$ is $2d$ we obtain $tr(A^{q})=n(B_{d}^{q})_{(0,0)}$ for $q=0,1,...,2d-1.$

Theorem 3.3

Let $\theta$ be a root of $H_{d-1}(x)-\epsilon.$ The multiplicity $m(\theta)$ of $\theta$ in $G$, $\theta\neq\pm k$, is given by

[TABLE]

Proof. In order to compute the multiplicity of an eigenvalue $\theta$ of $G$, we employ the same approach from [2], [4] and [13]. Let $\xi(x)=(x^{2}-k^{2})(H_{d-1}(x)+\frac{e}{2})(H_{d-1}(x)-1)$ and $\xi_{\theta}(x)=\frac{\xi(x)}{x-\theta}.$ Since $\xi(A)=0,$ it follows $m(\theta)=\frac{tr(\xi_{\theta}(A))}{\xi_{\theta}(\theta)}.$

As $deg(H_{d-1}(x))=d-1$ we obtain that $deg(\xi_{\theta}(x))=2d-1$. Therefore, let us assume $\xi_{\theta}(x)=x^{2d-1}+a_{2d-2}x^{2d-2}+...+a_{1}x+a_{0}.$ Hence,

[TABLE]

Since $tr(A^{q})=n(B_{d}^{q})_{0,0}$ for $0\leq q\leq 2d-1$, we have

[TABLE]

The polynomial $(x^{2}-k^{2})H_{d-1}(x)$ is a minimal polynomial of $B_{d}$, (see [15]). It implies

[TABLE]

Setting $L_{i+1}(x)=\frac{x^{2}-k^{2}}{x-\theta}(H_{i}(x)-H_{i}(\theta))$ for $i=0,...,d-1$, we get

[TABLE]

Therefore, $\xi_{\theta}(B_{d})=\frac{e}{2\epsilon}L_{d}(B_{d}).$

Calculating the derivative of $(x-\theta)\xi_{\theta}(x)$, that is, $((x-\theta)\xi_{\theta}(x))^{{}^{\prime}}=((x^{2}-k^{2})(H_{d-1}(x)+\frac{e}{2})(H_{d-1}(x)-1))^{{}^{\prime}}$, we have $\xi_{\theta}(\theta)=(2\epsilon+\frac{e}{2}-1)H_{d-1}^{{}^{\prime}}(\theta)(\theta^{2}-k^{2}).$ Thus

[TABLE]

In [13] was proven that $(L_{d}(B_{d}))_{0,0}=-k(k-1)H_{d-2}(\theta).$ Substituting it in the previous expression we obtain

[TABLE]

3.1 Multiplicities as function of $\cos\phi$

Let $\theta$ be a root of $H_{d-1}(x)-\epsilon$ and let $\theta=-2s\cos\phi,0<\phi<\pi$. We express the multiplicity of $\theta$, $m(\theta)$, as a function of $\cos\phi$. For that purpose we define the following functions $f(z),g_{1}(z),g_{2}(z)$ and $g_{3}(z).$

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Lemma 3.4

For either value of $\epsilon$, if we set $\theta_{i}=-2s\cos\phi_{i}$ for $1\leq i\leq d-1$, then

[TABLE]

[TABLE]

[TABLE]

Proof. The derivative of $H_{d-1}(x)$ is computed in [13]. We have

[TABLE]

Substituting $H_{d-2}(\theta_{i})=(-s)^{d-2}(-1)^{i+1}\frac{\sin(\phi_{i}+\alpha_{i})}{\sin\phi_{i}}$ and $H_{d-1}^{{}^{\prime}}(\theta_{i})$ in (5), we obtain

[TABLE]

The equation (4) yields $\sin(\phi_{i}+\alpha_{i})=\sin\phi_{i}(\cos\alpha_{i}+\eta_{i}s^{-d+1}\cos\phi_{i})$. Hence

[TABLE]

By equation (4) and Lemma 3.2, as $k,d\geq 3$, it follows that if $\eta_{i}=1$ or $\eta_{i}=\frac{e}{2}$ then $0<\alpha_{i}<\frac{\pi}{2}.$ Similarly, if $\eta_{i}=-1$ or $\eta_{i}=-\frac{e}{2}$, then $-\frac{\pi}{2}<\alpha_{i}<0.$ Therefore $\cos\alpha_{i}>0$, and thus, $\cos\alpha_{i}=\sqrt{1-\eta_{i}^{2}s^{-2d+2}(1-\cos^{2}\phi_{i})}$. It implies

[TABLE]

Using the formulas for $f,g_{1},g_{2}$ and $g_{3}$ we get the desired result.

The following two lemmas concern the monotonicity of $f,g_{1},g_{2}$ and $g_{3}.$ The first lemma is given in [4] and [13] (Lemma 3.5 and Lemma 4.1).

Lemma 3.5

For $k\geq 3$ and $\mid\!\!z\!\!\mid<1$ the function $f(z)$ is even and concave down.

Lemma 3.6

For $k\geq 3$, $d\geq 3$ and $\mid\!\!z\!\!\mid<1$, the monotonicity of $g_{1}(z),g_{2}(z)$ and $g_{3}(z)$ behave as follows.

• (1)

$g_{1}(z)$* is monotonic increasing;*

• (2)

$g_{2}(z)$* is monotonic decreasing;*

• (3)

$g_{3}(z)$* is monotonic increasing;*

Proof.

• (1)

It is suffice to prove that $g_{1}^{{}^{\prime}}(z)$ is positive on the interval $(-1,1)$. We have

[TABLE]

• (2)

In this case we prove that $g_{2}^{{}^{\prime}}(z)$ is negative on the interval $(-1,1)$.

[TABLE]

Since $k,d\geq 3$ and $k\geq e+2$, we easily conclude that $\frac{e^{2}}{4}s^{-2d+2}<1$ and $\mid\!\!\frac{e^{2}}{4}s^{-2d+2}(-1+z^{2})\!\!\mid<1$.

• (3)

It follows from the same reasoning as (2).

4 Main result

Let $\lambda_{1}<\lambda_{2}<...<\lambda_{d-1}$ be the roots of $H_{d-1}(x)+\frac{e}{2}$, and let $\mu_{1}<\mu_{2}<...<\mu_{d-1}$ be the roots of $H_{d-1}(x)-1.$

Lemma 4.1

Let $\lambda_{1},...,\lambda_{d-1}$ and $\mu_{1},...,\mu_{d-1}$ be defined as above. If $k\geq 3$ and $d\geq 3$ is an odd number, then

• (1)

$m(\lambda_{i})=m(\lambda_{d-i})$* and $m(\mu_{i})=m(\mu_{d-i})$, for $1\leq i\leq d-1$;*

• (2)

$m(\mu_{2})<m(\mu_{i})$* for $3\leq i\leq d-3$, and $m(\lambda_{1})<m(\lambda_{i})$, for $2\leq i\leq d-2.$*

Proof.

• (1)

If $d$ is odd $H_{d-1}(-x)=H_{d-1}(x)$. Therefore $\theta$ is a root of $H_{d-1}(x)-\epsilon$, if and only if, $-\theta$ is a root of $H_{d-1}(x)-\epsilon,$ (see [1]). Then $\lambda_{i}+\lambda_{d-i}=\mu_{i}+\mu_{d-i}=0$. By checking (5) and using $H_{d-2}(-x)=-H_{d-2}(x)$, we obtain $m(\lambda_{i})=m(\lambda_{d-i})$ and $m(\mu_{i})=m(\mu_{d-i})$ for each $1\leq i\leq d-1$.

• (2)

Since $\mu_{i}$ is a root of $H_{d-1}(x)-1$, we have $\epsilon=1.$ According to Lemma 3.2 let us set $\mu_{i}=-2s\cos\phi_{i}$, for $1\leq i\leq d-1$. In this case $\eta_{i}=\epsilon(-1)^{d+i}=(-1)^{i+1}.$ Since $-\mu_{2}=\mu_{d-2}$ we obtain $-\cos\phi_{2}=\cos\phi_{d-2}.$ Now, for $3\leq i\leq d-3$, we have $-\cos\phi_{2}=\cos\phi_{d-2}<\cos\phi_{i}<\cos\phi_{2}$. Since $f$ is even and concave down function we have

[TABLE]

The inequality $\cos\phi_{i}<|\cos\phi_{2}|$ and the fact that $g_{1}(z)$ is a monotonic increasing function yield $g_{1}(\eta_{2}\cos\phi_{2})=g_{1}(-\cos\phi_{2})<g_{1}(\pm\cos\phi_{i})$.

Therefore, for $3\leq i\leq d-3$, we conclude

[TABLE]

We proceed similarly when $\lambda_{i}$ is a root of $H_{d-1}(x)+\frac{e}{2}$. In this case $\epsilon=-\frac{e}{2}$ and $\eta_{i}=\frac{e}{2}(-1)^{i}.$ Again let $\lambda_{i}=-2s\cos\phi_{i},$ for $1\leq i\leq d-1$. Following the same reasoning as above we have $f(\cos\phi_{1})<f(\cos\phi_{i})$ for $2\leq i\leq d-2$.

Now, let $i$ be an odd number such that $3\leq i\leq d-2$. For such $i$ we note $\eta_{i}=-\frac{e}{2}<0$. Hence $g_{2}(z)$ is a monotonic decreasing, and therefore, $\cos\phi_{i}<\cos\phi_{1}$ yields $g_{2}(\cos\phi_{1})<g_{2}(\cos\phi_{i}).$ Thus, for odd $i$ such that $3\leq i\leq d-2$ we have

[TABLE]

Since $m(\lambda_{i})=m(\lambda_{d-i})$ occurs $m(\lambda_{1})<m(\lambda_{i})$ for each $2\leq i\leq d-2.$

Based on Lemma 4.1, we are ready to give the main result in this paper.

Theorem 4.2

Let $k\geq e+2\geq 4$ and $g=2d\geq 14$ be fixed. There exists no antipodal $(k,g)$-cage of excess $e$.

Proof. Since $m(\mu_{2})<m(\mu_{i})$ for $3\leq i\leq d-3$ and $\mu_{1}+\mu_{d-1}=0$, we obtain that $\mu_{2}$ and $\mu_{d-2}=-\mu_{2}$ are either conjugate quadratic irrationals or integers. Therefore, $\mu_{2}^{2}$ is an integer. Analogously, $\lambda_{2}^{2}$ is an integer. Hence $\lambda_{2}^{2}-\mu_{2}^{2}$ is an integer number. By Lemma 3.2 we have

[TABLE]

[TABLE]

Then, as $\lambda_{2}^{2}>4s^{2}\cos^{2}\frac{2\pi}{d}$ and $\mu_{2}^{2}<4s^{2}\cos^{2}\frac{2\pi}{d}$, we have that $\lambda_{2}^{2}-\mu_{2}^{2}>0.$ Furthermore, as $\lambda_{2}^{2}<4s^{2}\cos^{2}\frac{2\pi}{d+\frac{e}{2}s^{-d+1}}$ and $\mu_{2}^{2}>4s^{2}\cos^{2}\frac{2\pi}{d-s^{-d+1}}$, we have that

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Since $(d-s^{-d+1})^{2}>(d-1)^{2}>2d+1>2d+(\frac{e}{2}-1)s^{-d+1},$ it is suffices to prove that $d+\frac{e}{2}s^{-d+1}>4\pi\sqrt{\frac{e}{2}+1}s^{\frac{-d+3}{2}}.$ Using $k\geq e+2\geq 4$ and $d\geq 7$, we obtain

[TABLE]