Full Automorphism Group of (m, 2)-Graph in Finite Classical Polar Spaces

Abstract

Let $\mathcal{Q}$ be the finite classical polar space of rank $\nu \ge 1$ over ${\mathbb{F}}_q$, and ${\mathcal{Q}}_m$ be the set of all m-dimensional subspaces of $\mathcal{Q}$. In this paper, we introduce the (m, 2)-graph with ${\mathcal{Q}}_m$ as its vertex set, and two vertices P, Q are adjacent if and only if P + Q is an (m + 2)-dimensional subspace of $\mathcal{Q}$. The full automorphism group of (m, 2)-graph is determined.

1. Introduction

Finite classical polar spaces arise from the classification of non-degenerate sesquilinear and nonsingular quadratic forms defined on vector spaces over a finite field. There are six fundamental types of forms to consider, which naturally induce three geometric structures. Alternating forms yield symplectic geometries, Hermitian forms generate unitary geometries, and quadratic forms produce orthogonal geometries. Given a form on a finite vector space V, all totally isotropic subspaces of V form the associated polar space $\mathcal{Q}$. The rank of the polar space $\mathcal{Q}$ is defined as the dimension of its maximal subspaces. The collection of all m-dimensional subspaces of $\mathcal{Q}$ is denoted by ${\mathcal{Q}}_m$.

Table 1. Finite classical polar spaces of rank $\nu$.

Name	Symbol	n	$\mathit{ϵ}$	G
symplectic	$W_{2\nu -1}\left(q\right)$	$2\nu$	0	$P\Gamma S{p}_{2\nu }\left(q\right)$
Hermitian	$U_{2\nu -1}\left(q\right)$	$2\nu$	$-1/2$	$P\Gamma U_{2\nu }\left(\sqrt{q}\right)$
Hermitian	$U_{2\nu }\left(q\right)$	$2\nu +1$	$1/2$	$P\Gamma U_{2\nu +1}\left(\sqrt{q}\right)$
hyperbolic	$Q_{2\nu -1}^{+}\left(q\right)$	$2\nu$	$-1$	$P\Gamma O_{2\nu }^{+}\left(q\right)$
parabolic	$Q_{2\nu }\left(q\right)$	$2\nu +1$	0	$P\Gamma O_{2\nu +1}\left(q\right)$
elliptic	$Q_{2\nu +1}^{-}\left(q\right)$	$2\nu +2$	1	$P\Gamma O_{2\nu +2}^{-}\left(q\right)$

classifies all isomorphism types of finite classical polar spaces with rank $\nu \ge 1$ over finite fields ${\mathbb{F}}_q$. The table specifies type symbols for each polar space class, dimension n of the corresponding projective space $PG\left(V\right)$, a uniformity parameter $ϵ$ facilitating consistent enumeration across types, and the collineation group G. In particular, the Hermitian polar spaces only exist for squares q. For more details on finite classical polar spaces, readers can refer to [1,2,3].

The $(m,t)$-graph in finite classical polar spaces $\mathcal{Q}$ is the graph with ${\mathcal{Q}}_m$ as its vertex set, and two vertices $P,Q$ are adjacent if and only if $P+Q$ is an $(m+t)$-dimensional subspace of $\mathcal{Q}$. It is easy to verify that the union of $(m,1)$-graph and generalized symplectic, unitary or orthogonal graph is a Grassmann graph of the corresponding polar spaces. The properties and automorphism groups of generalized symplectic, unitary, and orthogonal graphs have been extensively studied by many scholars [4,5,6,7,8,9,10,11,12,13]. In this paper, we determine the full automorphism group of the $(m,2)$-graph. For convenience, we denote the $(m,2)$-graph in $\mathcal{Q}$ by $\Gamma$. There is an edge in graph $\Gamma$ if and only if $2\le m\le \nu -2$. Therefore, we only consider these cases. The main theorems are as follows.

Theorem 1.

The graph Γ is a $(v,k)$-regular graph where

$$v={\displaystyle \frac{{\displaystyle \prod _{i=\nu -m+1}^{\nu }}(q^i-1)(q^{i+ϵ}+1)}{{\displaystyle \prod _{i=1}^m}(q^i-1)}},$$

and

$$k={\displaystyle \frac{q^4{\displaystyle \prod _{i=m-1}^m}(q^i-1){\displaystyle \prod _{i=\nu -m-1}^{\nu -m}}(q^i-1)(q^{i+ϵ}+1)}{{(q-1)}^2{(q^2-1)}^2}}.$$

Theorem 2.

Let $2\le m\le \nu -2$; the full automorphism group of the graph Γ is the collineation group G in Table 1, that is, every automorphism of Γ has the following form:

$$\begin{array}{cc}\hfill {\sigma }_{\pi ,T}:V(\Gamma )& \to V(\Gamma )\hfill \\ \hfill X& \mapsto \pi \left(X\right)T,\hfill \end{array}$$

where $\pi \in \mathrm{Aut}\left({\mathbb{F}}_q\right)$, $\pi \left(X\right)=\left(\pi \left(x_{ij}\right)\right)$, and T belongs to the projective generalized classical group of the corresponding polar space.

In the following sections of this paper, we will take the symplectic polar space as an example and give proofs of Theorems 1 and 2. The cases on unitary and orthogonal polar spaces are similar. Section 2 introduces the preliminaries on the symplectic polar space. In Section 3, we investigate the properties of the $(m,2)$-graph in the symplectic polar space. Section 4 determines the full automorphism group of the $(m,2)$-graph. Finally, in Section 5, we summarize our work and propose two open questions for further research.

2. Preliminaries on the Symplectic Polar Space

Let K be an $n\times n$ nonsingular alternate matrix over ${\mathbb{F}}_q$. As we know, n is even. From now on, we assume that $n=2\nu$. A $2v\times 2\nu$ matrix T over ${\mathbb{F}}_q$ is called a symplectic matrix with respect to K if $TK{T}^T=K$, where $T^T$ is the transpose of the matrix T. Clearly, $2v\times 2\nu$ symplectic matrices form a group with respect to the matrix multiplication, called the symplectic group of degree $2\nu$ with respect to K over ${\mathbb{F}}_q$ and denoted by $S{p}_{2\nu }({\mathbb{F}}_q,K)$. It is easy to verify that the symplectic groups of degree $2\nu$ with respect to different $2v\times 2\nu$ nonsingular alternate matrices are isomorphic. Thus, in discussing symplectic groups, without loss of generality, we can choose any particular $2v\times 2\nu$ nonsingular alternate matrix K. In this paper, let us take

$$K=\left(\begin{array}{cc}0& I^{\left(\nu \right)}\\ -I^{\left(\nu \right)}& 0\end{array}\right),$$

where $I^{\left(\nu \right)}$ is the $\nu \times \nu$ identity matrix. Denote the symplectic group with respect to K simply by $S{p}_{2\nu }\left({\mathbb{F}}_q\right)$. Consider the $2\nu$-dimensional row vector space ${\mathbb{F}}_q^{\left(2\nu \right)}$ over the finite field ${\mathbb{F}}_q$. The symplectic group $S{p}_{2\nu }\left({\mathbb{F}}_q\right)$ acts naturally on this space via right matrix multiplication, as expressed by the group action map:

$$\begin{array}{c}\hfill \begin{array}{ccc}\hfill {\mathbb{F}}_q^{\left(2\nu \right)}\times S{p}_{2\nu }\left({\mathbb{F}}_q\right)& ⟶& {\mathbb{F}}_q^{\left(2\nu \right)}\hfill \\ \hfill (\alpha ,T)& ⟼& \alpha T.\hfill \end{array}\end{array}$$

The vector space ${\mathbb{F}}_q^{\left(2\nu \right)}$, together with the above group action, forms the $2\nu$-dimensional symplectic space over ${\mathbb{F}}_q$.

Let P denote an m-dimensional subspace of ${\mathbb{F}}_q^{\left(2\nu \right)}$. A matrix representation of P is any $m\times 2\nu$ matrix whose row space spans P. Note that such representations are unique up to left multiplication by invertible matrices. Two $m\times 2\nu$ matrices $P_1$ and $P_2$ of rank m represent the same subspace if and only if $P_1=Q{P}_2$ for some $Q\in G{L}_m\left({\mathbb{F}}_q\right)$, denoted by $P_1\doteq P_2$. The m-dimensional subspace P is classified as type $(m,s)$ if the rank of $PK{P}^T$ is $2s$, that is, rk$\left(PK{P}^T\right)=2s$. The existence of such subspaces is constrained by the inequalities $2s\le m\le \nu +s$ [14]. Notably, when $s=0,$ i.e., $PK{P}^T=0$, P is called totally isotropic. The collection of all totally isotropic subspaces constitutes the symplectic polar space of rank $\nu$, denoted by $W_{2\nu -1}\left(q\right)$. For brevity, we still adopt $\mathcal{Q}$ for $W_{2\nu -1}\left(q\right)$, ${\mathcal{Q}}_m$ for the set of m-dimensional totally isotropic subspaces, and $dimP$ for the dimension of P.

Lemma 1

([14]). $\left(i\right)$ Let $1\le m\le n$. Then the number of m-dimensional vector subspaces of ${\mathbb{F}}_q^{\left(n\right)}$ is

$$N(m,n)={\displaystyle \frac{{\prod }_{i=n-m+1}^n(q^i-1)}{{\prod }_{i=1}^m(q^i-1)}}.$$

(ii) The number of subspaces in ${\mathcal{Q}}_m$ is equal to

$$N(m,0,2\nu )={\displaystyle \frac{{\displaystyle \prod _{i=\nu -m+1}^{\nu }}(q^{2i}-1)}{{\displaystyle \prod _{i=1}^m}(q^i-1)}}.$$

Lemma 2

([15]). $\left(i\right)$ Let $X,Y\in {\mathcal{Q}}_m$, $dim(X\cap Y)=d$ and $\mathrm{rk}\left(XK{Y}^T\right)=r$. Then $0\le d\le m-1$ and

$$max\{0,2m-\nu -d\}\le r\le m-d.$$

$\left(ii\right)$ Let $X_1,Y_1,X_2,Y_2\in {\mathcal{Q}}_m$. Then $dim(X_1\cap Y_1)=dim(X_2\cap Y_2)$ and $\mathrm{rk}\left(X_{1}K{Y}_1^T\right)=\mathrm{rk}\left(X_{2}K{Y}_2^T\right)$ if and only if there exists $T\in S{p}_{2\nu }\left({\mathbb{F}}_q\right)$ such that $X_2=X_{1}T$ and $Y_2=Y_{1}T$.

Let $P\in {\mathcal{Q}}_m$; the number of vertices in the set

$$\{Q\in V(\Gamma )\mid dim(P\cap Q)=d,\mathrm{rk}\left(PK{Q}^T\right)=r\}$$

is independent of the choice of P, since the symplectic group acts transitively on the subspaces of the same type. Denote this set by $S(d,r)$.

Lemma 3

([15]). Let $1\le m\le \nu ,0\le d\le m-1,max\{0,2m-\nu -d\}\le r\le m-d.$ Then the number of vertices in $S(d,r)$ is

$$n(d,r)=N(d,m)N(r,m-d)N(m-d-r,0;2(\nu -m))q^{2r(\nu -m)+{(m-d-r)}^2+{\displaystyle \frac{r(r+1)}{2}}}.$$

The generalized symplectic graph $GS{p}_{2\nu }(q,m)$ over ${\mathbb{F}}_q$ is the graph with ${\mathcal{Q}}_m$ as its vertex set, and two vertices P and Q are adjacent if and only if

$$dim(P\cap Q)=m-1\mathrm{and}\mathrm{rk}\left(PK{Q}^T\right)=1.$$

To introduce the full automorphism group of $GS{p}_{2\nu }(q,m)$, we give the definition of the projective generalized symplectic group. The set of $2\nu \times 2\nu$ matrices T over the finite field ${\mathbb{F}}_q$ satisfying the condition $TK{T}^T=kK$, where $k\in {\mathbb{F}}_q^{*}$, forms the generalized symplectic group of degree $2\nu$ over ${\mathbb{F}}_q$ with respect to the matrix multiplication, denoted by $GS{p}_{2\nu }\left({\mathbb{F}}_q\right).$ The projective quotient $PGS{p}_{2\nu }\left({\mathbb{F}}_q\right)$ is obtained by factoring the $GS{p}_{2\nu }\left({\mathbb{F}}_q\right)$ modulo its central subgroup $\left\{k{I}^{\left(2\nu \right)}\right|k\in {\mathbb{F}}_q^{*}\}$, called the projective generalized symplectic group.

Lemma 4

([6]). The full automorphism group of the generalized symplectic graph $GS{p}_{2\nu }(q,m)$ is $P\Gamma S{p}_{2\nu }\left(q\right)$, that is, every automorphism of $GS{p}_{2\nu }(q,m)$ has the following form:

$$\begin{array}{cc}\hfill {\sigma }_{\pi ,T}:V\left(GS{p}_{2\nu }(q,m)\right)& \to V\left(GS{p}_{2\nu }(q,m)\right)\hfill \\ \hfill X& \mapsto \pi \left(X\right)T,\hfill \end{array}$$

where $\pi \in \mathrm{Aut}\left({\mathbb{F}}_q\right)$, $\pi \left(X\right)=\left(\pi \left(x_{ij}\right)\right)$, and $T\in PGS{p}_{2\nu }\left({\mathbb{F}}_q\right)$.

3. Properties of $(\mathit{m},\mathbf{2})$-Graph in the Symplectic Polar Space

The $(m,2)$-graph in the symplectic polar space $\mathcal{Q}$ is the graph $\Gamma$ with ${\mathcal{Q}}_m$ as its vertex set, and two vertices $P,Q$ are adjacent if and only if $P+Q$ is an $(m+2)$-dimensional totally isotropic subspace. From the conditions for the existence of an $(m,s)$ subspace, it follows that there is an edge in the graph $\Gamma$ if and only if $2\le m\le \nu -2$.

Since the symplectic group acts transitively on the subspaces of the same type, the graph $\Gamma$ is vertex-transitive. It is obvious that the number of vertices in $\Gamma$ is $N(m,0,2\nu )$ in Lemma 1, and the degree of each vertex is $n(m-2,0)$ in Lemma 3. By simple calculation, Theorem 1 is proved, where $ϵ=0$.

Let $\partial (P,Q)$ be the distance between two vertices P and Q in $\Gamma$. Denote by $P\sim Q$ when P and Q are adjacent, and $P\nsim Q$ otherwise. For $P\in V(\Gamma )$, let $\Gamma \left(P\right)$ be the set of neighbors of P. Then we have the following theorem.

Theorem 3.

Let $2\le m\le \nu -2$, $P,Q$ be two vertices of Γ satisfying $dim(P\cap Q)=d$ and $\mathrm{rk}\left(PK{Q}^T\right)=r$; then, $\partial (P,Q)=2$ if and only if

$$m-4\le d\le m-1,max\{0,2m-\nu -d\}\le r\le min\{2,m-d\},$$

where $r\ne 0$ when $d=m-2$.

Proof. 

Since the symplectic group is an automorphism group of $\Gamma$, without loss of generality, we can assume that

where the numbers above the matrix are the numbers of columns. If $dim(P\cap Q)=d$ and $\mathrm{rk}\left(PK{Q}^T\right)=r$, then $0\le d\le m-1$ and

$$max\{0,2m-\nu -d\}\le r\le m-d$$

by Lemma 1, and we can choose a special vertex Q to compute its distance with P by Lemma 2. Next, let us discuss d with the following cases.

$\left(i\right)$ If $d=m-1$, then $0\le r\le 1$ by $2\le m\le \nu -2$ and

$$max\{0,2m-\nu -d\}\le r\le min\{2,m-d\}.$$

When $r=0$, we can assume that

When $r=1$, we can assume that

, that is, $\partial (P,Q)\ge 2$. Let

It is to easy to verify that $P\sim R\sim Q$; thus, $\partial (P,Q)=2$.

$\left(ii\right)$ If $d=m-2$, then $0\le r\le 2$.

When $r=0$, $P\sim Q$, that is, $\partial (P,Q)=1$.

When $r=1$ or 2, similar to the discussion in case (i), we could find a vertex R satisfying $P\sim R\sim Q$ for each special Q. Thus, $\partial (P,Q)=2$.

$\left(iii\right)$ If $d=m-3$, then $max\{0,m+3-\nu \}\le r\le 3$ (this case occurs only when $m\ge 3$).

When $r=0,1$ or 2, similar to the discussion in case (i), we have $\partial (P,Q)=2$.

When $r=3$, we can assume

For each $M\in \Gamma \left(P\right)$, it has the following matrix representation

where rk$(A_{11},A_{12})=m-2$, and $(B_2,D_2)\in {\mathcal{Q}}_2.$ Thus,

Since rk$(A_{11},A_{12})=m-2$ and $A_{12}\in {\mathbb{F}}_q^{(m-2)\times (m-3)}$, rk$\left(A_{11}\right)\ge 1$. This means that $0\le dim(M\cap Q)\le m-3$. Therefore, $\partial (P,Q)>2$. In addition, let

where $e_1$ and $e_2$ are the first and second unit vectors, respectively. It is easy to verify that $P\sim R\sim S\sim Q$, then $\partial (P,Q)=3.$

$\left(iv\right)$ If $d=m-4$, then $max\{0,m+4-\nu \}\le r\le 4$ (this case occurs only when $m\ge 4$).

Similar to the discussion in case (iii), we have $\partial (P,Q)=2$ when $r=0,1$ or 2, and $\partial (P,Q)=3$ when $r=3$ or 4.

$\left(iv\right)$ If $0\le d\le m-5$, then $max\{0,m+5-\nu \}\le r\le 5$ (this case occurs only when $m\ge 5$).

Similar to the discussion in case (iii), we have $\partial (P,Q)>2$.

Above all, $\partial (P,Q)=2$ if and only if $d=m-1$, $d=m-2$, or $0\le r\le 2$ when $d=m-3$ or $m-4$. The theorem now follows. □

From the proof of the above theorem, we have the following corollary.

Corallary 1.

Let the diameter of Γ be $d(\Gamma )$. Then, $d(\Gamma )=2$ when $m=2$, and $d(\Gamma )=3$ when $m=3$ or 4.

4. Automorphisms of $(\mathit{m},\mathbf{2})$-Graph in the Symplectic Polar Space

In this section, we will determine the full automorphism group of the $(m,2)$-graph $\Gamma$ in the symplectic polar space. For the given d and r, let P and Q be any two vertices of $\Gamma$ satisfying $dim(P\cap Q)=d,\mathrm{rk}\left(PK{Q}^T\right)=r.$ Then, by Lemma 2, the number of common neighbors of P and Q is independent of the choice of P and Q, which is denoted by $l(d,r)$. We have the following theorem.

Theorem 4.

Let $2\le m\le \nu -2,m-4\le d\le m-1,$

$$max\{0,2m-\nu -d\}\le r\le min\{2,m-d\},$$

where $r\ne 0$ when $d=m-2$. Then,

$\left(i\right)$$l(m-1,0)\ne l(m-1,1)$.

$\left(ii\right)$$l(m-1,t)\ne l(d,r),$ where $t=0$ or 1, $d<m-1$.

Proof. 

The conditions that d and r satisfy are precisely those stated in Theorem 3. Without loss of generality, let

If $dim(P\cap Q)=d$ and $\mathrm{rk}\left(PK{Q}^T\right)=r$ as in the theorem, then $\partial (P,Q)=2$. Suppose $M\in \Gamma \left(P\right)$, then M has the matrix representation

where

$$\mathrm{rk}(M_{11},M_{12})=m-2,\left(\begin{array}{cc}{\beta }_1& {\gamma }_1\\ {\beta }_2& {\gamma }_2\end{array}\right)\in {\mathcal{Q}}_2.$$

$l(d,r)$ is the number of common neighbors of P and Q, i.e., the number of vertices M such that $Q\sim M$. Next, we will compute $l(d,r)$ by choosing the special Q for each pair $(d,r)$.

When $(d,r)=(m-1,1)$, let

$l(m-1,1)$ is the number of M satisfying $Q\sim M$. $Q\sim M$ if and only if

This means that $M_{11}=0,a_1=a_2=0$, and then

where $\mathrm{rk}\left(M_{12}\right)=m-2$. Therefore,

$$\begin{array}{cc}\hfill l(m-1,1)=& q^{2}N(m-2,m-1)N(2,0;2(\nu -m))\hfill \\ \hfill =& {\displaystyle \frac{q^2(q^{m-1}-1)(q^{2(\nu -m-1)}-1)(q^{2(\nu -m)}-1)}{{(q-1)}^2(q^2-1)}}.\hfill \end{array}$$

Through similar computations, we could get

$$\begin{array}{cc}\hfill l(m-1,0)=& {\displaystyle \frac{q^3(q^{m-1}-1)(q^{2(\nu -m-1)}-1)}{q-1}}+{\displaystyle \frac{q^6(q^{m-1}-1)(q^{m-2}-1)(q^{2(\nu -m-1)}-1)}{{(q-1)}^2(q^2-1)}}\hfill \\ \hfill +& {\displaystyle \frac{q^6(q^{m-1}-1)(q^{2(\nu -m-2)}-1)(q^{2(\nu -m-1)}-1)}{{(q-1)}^2(q^2-1)}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill l(m-2,1)=& q(q^{2(\nu -m-1)}-1)+{\displaystyle \frac{q^3(q^{m-2}-1)(q^{2(\nu -m-1)}-1)}{{(q-1)}^2}}\hfill \\ & +{\displaystyle \frac{q^4(q^{2(\nu -m-2)}-1)(q^{2(\nu -m-1)}-1)}{(q-1)(q^2-1)}}.\hfill \end{array}$$

$$\begin{array}{c}\hfill l(m-2,2)={\displaystyle \frac{(q^{2(\nu -m-1)}-1)(q^{2(\nu -m)}-1)}{(q-1)(q^2-1)}}.\end{array}$$

$$\begin{array}{cc}\hfill l(m-3,0)=& {(q^2+q+1)}^2(q-1){(q+1)}^2+{\displaystyle \frac{(q^{m-3}-1)}{q-1}}(q^2+q+1)(q^6+q^5+q^4)\hfill \\ \hfill +& {\displaystyle \frac{q^4{(q^2+q+1)}^2(q^{2(\nu -m-3)}-1)}{(q-1)}}.\hfill \end{array}$$

$$\begin{array}{cc}\hfill l(m-3,1)=& {\displaystyle \frac{q^4(q^{m-3}-1)}{q-1}}+{(q+1)}^2(q-1)+{\displaystyle \frac{{(q^2+q)}^2(q^{2(\nu -m-2)}-1)}{(q-1)}}.\hfill \end{array}$$

$$\begin{array}{c}\hfill l(m-3,2)={\displaystyle \frac{q^{2(\nu -m-1)}-1}{(q-1)}},l(m-4,0)={(q^2+q+1)}^2{(1+q^2)}^2.\end{array}$$

$$\begin{array}{c}\hfill l(m-4,1)={(q^2+q+1)}^2,l(m-4,2)=1.\end{array}$$

Above all, we can verify that $q^3\mid l(m-1,0)$, $q^4\nmid l(m-1,0)$, $q^2\mid l(m-1,1)$, $q^3\nmid l(m-1,1)$, $q^2\nmid l(d,r)$ for $d<m-1$. Then, the theorem follows. □

Finally, we present the proof of Theorem 2.

Proof of Theorem 2.

It is easy to verify that the group $P\Gamma S{p}_{2\nu }\left(q\right)$ is an automorphism group of $\Gamma$. Conversely, let $\sigma$ be an automorphism of $\Gamma$, and $P,Q\in V(\Gamma )$ satisfies $dim(P\cap Q)=m-1,\mathrm{rk}\left(PK{Q}^T\right)=1.$ Then, $\partial (P,Q)=2$ by Theorem 3. Since $\sigma$ preserves the distance between two vertices in $\Gamma$, $\partial \left(\sigma \right(P),\sigma (Q\left)\right)=2$, and

$$|\Gamma (P)\cap \Gamma (Q\left)\right|=|\Gamma (\sigma \left(P\right))\cap \Gamma (\sigma \left(Q\right)\left)\right|.$$

Therefore, $dim(\sigma \left(P\right)\cap \sigma \left(Q\right))=m-1,\mathrm{rk}\left(\sigma \left(P\right)K\sigma {\left(Q\right)}^T\right)=1$ by Theorem 4, which means that $\sigma$ is also an automorphism of the generalized symplectic graph $GS{p}_{2\nu }(q,m)$. Thus, $\sigma \in P\Gamma S{p}_{2\nu }\left(q\right)$ by Lemma 4. The theorem now follows. □

5. Conclusions and Further Study

In this paper, we introduce the $(m,t)$-graph in finite classical polar spaces. Taking the symplectic polar space as an example, we establish fundamental properties of the $(m,2)$-graph $\Gamma$, proving its regularity and characterizing distance metric, completely determining its full automorphism group. In the following study, we will consider the spectral properties of $\Gamma$, and examine whether similar results extend to the general $(m,t)$-graph with $t\ge 3$. These investigations will deepen the connections between finite geometry, algebraic graph theory, and group theory.