Generalized bilinear forms graphs and MDR codes

TL;DR

This paper studies the properties of generalized bilinear forms graphs over residue class rings, revealing their structure, maximum cliques, and their connection to maximum rank distance (MRD) and maximum distance separable (MDS) codes.

Contribution

It characterizes the graph-theoretic properties of $ ext{Gamma}_d$ over $ ext{Z}_{p^s}$ and links largest independent sets to optimal error-correcting codes.

Findings

$ ext{Gamma}_d$ is connected and vertex transitive.

Largest independent sets are MRD and MDS codes.

Largest independent sets can be linear codes.

Abstract

We investigate the generalized bilinear forms graph $\Gamma_d$ over a residue class ring $\mathbb{Z}_{p^s}$. We show that $\Gamma_d$ is a connected vertex transitive graph, and completely determine its independence number, clique number, chromatic number and maximum cliques. We also prove that cores of both $\Gamma_d$ and its complement are maximum cliques. The graph $\Gamma_d$ is useful for error-correcting codes. We show that every largest independent set of $\Gamma_d$ is both an MRD code over $\mathbb{Z}_{p^s}$ and a usual MDS code. Moreover, there is a largest independent set of $\Gamma_d$ to be a linear code over $\mathbb{Z}_{p^s}$.

Equations

$$A=BC$$

$${\mathrm{d}}_{\mathrm{R}}(A,B)=\rho (A-B).$$

$$\left|{\mathbb{Z}}_{p^s}\right|=p^s,\left|{\mathbb{Z}}_{p^s}^{\ast }\right|=(p-1)p^{s-1},\left|J_{p^s}\right|=p^{s-1}.$$

$$x=t_0+t_{1}p+\cdots +t_{s-1}{p}^{s-1},$$

$$X=X_0+X_{1}p+\cdots +X_{s-1}{p}^{s-1},$$

$$\pi :{\mathbb{Z}}_{p^s}^{m\times n}\to {\mathbb{Z}}_p^{m\times n}$$

$$\pi (X+Y)=\pi (X)+\pi (Y),$$

$$\pi (XQ)=\pi (X)\pi (Q),$$

$$\pi {(}^{t}X)={}^t(\pi (X)).$$

$$A=P\mathrm{diag}\left(I_r,p^{k_1},\dots ,p^{k_t},0\right)Q,$$

$$\mathrm{rk}(A)=\max \left\{k:{\mathrm{Ann}}_R(I_k(A))=(0)\right\}.$$

$$\rho\left(\begin{array}[]{cc}A&0\\ 0&B\\ \end{array}\right)=\rho(A)+\rho(B)$$

$$\chi (G)\ge \max \left\{\omega (G),|V(G)|/\alpha (G)\right\}.$$

$$\chi(G)\geq\frac{|V(G)|}{\alpha(G)}\geq\omega(G),\ \mbox{if $G$ is vertex-transitive.}$$

$$\alpha \left({\Gamma }_d({\mathbb{F}}_q^{m\times n})\right)=q^{n(m-d+1)}$$

$$\omega \left({\Gamma }_d({\mathbb{F}}_q^{m\times n})\right)=q^{n(d-1)}$$

$$\alpha \left({\Gamma }_d({\mathbb{Z}}_{p^s}^{m\times n})\right)=p^{sn(m-d+1)}$$

$$\omega \left({\Gamma }_d({\mathbb{Z}}_{p^s}^{m\times n})\right)=p^{sn(d-1)}.$$

$$\alpha \left({\Gamma }_d({\mathbb{Z}}_{p^{s-1}}^{m\times n})\right)=p^{(s-1)n(m-d+1)}.$$

$${\mathcal{L}}_k=S_k+{\mathcal{A}}_{s-1}p=\left\{S_k+C_{1}p,S_k+C_{2}p,\dots ,S_k+C_{\beta }p\right\},k=1,\dots ,\alpha .$$

$${\mathcal{A}}_s={\mathcal{L}}_1\cup {\mathcal{L}}_2\cup \cdots \cup {\mathcal{L}}_{\alpha }:={\mathcal{A}}_1+{\mathcal{A}}_{s-1}p.$$

$$\alpha \left({\Gamma }_d({\mathbb{Z}}_{p^s}^{m\times n})\right)\ge p^{sn(m-d+1)}.$$

$$\omega \left({\Gamma }_d({\mathbb{Z}}_{p^s}^{m\times n})\right)\le \frac{\left|V\left({\Gamma }_d({\mathbb{Z}}_{p^s}^{m\times n}\right)\right|}{\alpha \left({\Gamma }_d({\mathbb{Z}}_{p^s}^{m\times n})\right)}\le \frac{p^{smn}}{p^{sn(m-d+1)}}=p^{sn(d-1)}.$$

$$\omega \left({\Gamma }_d({\mathbb{Z}}_{p^s}^{m\times n})\right)=p^{sn(d-1)}.$$

$$\alpha \left({\Gamma }_d({\mathbb{Z}}_{p^s}^{m\times n})\right)\le \frac{\left|V\left({\Gamma }_d({\mathbb{Z}}_{p^s}^{m\times n}\right)\right|}{\omega \left({\Gamma }_d({\mathbb{Z}}_{p^s}^{m\times n})\right)}=\frac{p^{smn}}{p^{sn(d-1)}}=p^{sn(m-d+1)}.$$

$$\mbox{$\chi\left(\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})\right)=\omega\left(\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})\right)=p^{sn(d-1)}$ (where $n\geq m$)}.$$

$$d\le n-({\log }_{q}M)+1.$$

$$W=(I_n,0)$$

$${\mathcal{A}}_t=\left\{U:U\subseteq {\mathbb{F}}_q^{m+n},\dim (U)=t,U\cap W=\{0\}\right\},t=m-1,m.$$

$$\phi (U)=X_U,U=(X_U,I_m)\in {\mathcal{A}}_m.$$

Full text

Keywords: bilinear forms graph, residue class ring, independence number, core, maximum clique, MDR code

2010 AMS Classification: 05C25, 15B33, 05C30, 94B60, 94B65

1 Introduction

Throughout, let $R$ be a commutative local ring and $R^{*}$ the set of all units of $R$. For a subset $S$ of $R$, let $S^{m\times n}$ be the set of all $m\times n$ matrices over $S$, and let $S^{n}=S^{1\times n}$. Let $GL_{n}(R)$ be the set of $n\times n$ invertible matrices over $R$. Let ${}^{t}A$ denote the transpose matrix of a matrix $A$. Denote by $I_{r}$ ($I$ for short) the $r\times r$ identity matrix, and ${\rm diag}(A_{1},\ldots,A_{k})$ a block diagonal matrix where $A_{i}$ is an $m_{i}\times n_{i}$ matrix. The cardinality of a set $X$ is denote by $|X|$.

For $0\neq A\in R^{m\times n}$, by Cohn’s definition [6], the inner rank of $A$, denoted by $\rho(A)$, is the least positive integer $r$ such that

[TABLE]

Let $\rho(0)=0$. For $A\in R^{m\times n}$, it is clear that $\rho(A)\leq\min\left\{m,n\right\}$ and $\rho(A)=0$ if and only if $A=0$. When $R$ is a field, we have $\rho(A)={\rm rank}(A)$, where ${\rm rank}(A)$ is the usual rank of matrix over a field. For matrices over $R$, we have (cf. [6, 5, Section 5.4]): $\rho(A)=\rho(PAQ)$ where $P$ and $Q$ are invertible matrices over $R$; $\rho(A+B)\leq\rho(A)+\rho(B)$ and $\rho(AC)\leq\min\left\{\rho(A),\rho(C)\right\}$.

For $A,B\in R^{m\times n}$, the rank distance between $A$ and $B$ is defined by

[TABLE]

We have that ${\rm d_{R}}(A,B)=0\Leftrightarrow A=B$, ${\rm d_{R}}(A,B)={\rm d_{R}}(B,A)$ and ${\rm d_{R}}(A,B)\leq{\rm d_{R}}(A,C)+{\rm d_{R}}(C,B)$, for all matrices of appropriate sizes $A,B,C$ over $R$.

Let $\mathbb{Z}_{p^{s}}$ denote the residue class ring of integers modulo $p^{s}$, where $p$ is a prime and $s$ is a positive integer. The $\mathbb{Z}_{p^{s}}$ is a Galois ring, a commutative local ring, a finite principal ideal ring (cf. [20, 27]). The principal ideal $(p)$ is the unique maximal ideal of $\mathbb{Z}_{p^{s}}$, and denoted by $J_{p^{s}}$. The $J_{p^{s}}$ is also the Jacbson radical of $\mathbb{Z}_{p^{s}}$. When $s=1$, $\mathbb{Z}_{p}$ is a finite field with $p$ elements. We have (cf. [20, 27]) that

[TABLE]

Let $\mathbb{F}_{q}$ be the finite field with $q$ elements (where $q$ is a power of a prime). All graphs are simple [14] and finite in this paper. Let $V(G)$ denote the vertex set of a graph $G$. For $x,y\in V(G)$, we write $x\sim y$ if vertices $x$ and $y$ are adjacent. Denote by ${\rm Aut}(G)$ the automorphism group of a graph $G$.

The generalized bilinear forms graph over $\mathbb{F}_{q}$, denoted by $\Gamma_{d}(\mathbb{F}_{q}^{m\times n})$, has the vertex set $\mathbb{F}_{q}^{m\times n}$ where $m,n\geq 2$, and two distinct vertices $A$ and $B$ are adjacent if ${\rm rank}(A-B)<d$ where $d$ is fixed with $2\leq d\leq{\rm min}\{m,n\}$. When $d=2$, $\Gamma_{2}(\mathbb{F}_{q}^{m\times n})$ is the usual bilinear forms graph over $\mathbb{F}_{q}$. The bilinear forms graph plays an important role in combinatorics and coding theory, and it has been extensively studied (cf. [3, 8, 12, 16, 26, 28]).

Recently, the bilinear forms graph over $\mathbb{Z}_{p^{s}}$ is studied by [17]. However, the generalized bilinear forms graph over $\mathbb{Z}_{p^{s}}$ remains to be further studied. As a natural extension of the generalized bilinear forms graph over $\mathbb{F}_{q}$, we define the generalized bilinear forms graph over $\mathbb{Z}_{p^{s}}$ as follows. The generalized bilinear forms graph (bilinear forms graph for short) over $\mathbb{Z}_{p^{s}}$, denoted by $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ ($\Gamma_{d}$ for short), has the vertex set $\mathbb{Z}_{p^{s}}^{m\times n}$ where $m,n\geq 2$, and two distinct vertices $A$ and $B$ are adjacent if $\rho(A-B)<d$, where $d$ is fixed with $2\leq d\leq{\rm min}\{m,n\}$.

MRD codes and codes over $\mathbb{Z}_{p^{s}}$ are active research topics in the coding theory (cf. [8, 11, 24, 7, 15, 22]) and [9, 10, 19]). The generalized bilinear forms graph $\Gamma_{d}$ has good application to the error-correcting codes over $\mathbb{Z}_{p^{s}}$. In fact, we will show that every largest independent set of $\Gamma_{d}$ is both an MRD code over $\mathbb{Z}_{p^{s}}$ and a usual MDS code [25]. Moreover, there is a largest independent set of $\Gamma_{d}$ such that it is a linear code over $\mathbb{Z}_{p^{s}}$.

The paper is organized as follows. In Section 2, we recall some properties of matrices over $\mathbb{Z}_{p^{s}}$. In Section 3, we show that $\Gamma_{d}$ is a connected vertex transitive graph, and determine the independence number, the clique number and the chromatic number of $\Gamma_{d}$. We also show that every largest independent set of $\Gamma_{d}$ is both an $(m\times n,d)$ MRD code over $\mathbb{Z}_{p^{s}}$ and a usual MDS code, and there is a largest independent set of $\Gamma_{d}$ to be a linear code over $\mathbb{Z}_{p^{s}}$. In Section 4, We will determine the algebraic structures of maximum cliques of $\Gamma_{d}$, and show that cores of both $\Gamma_{d}$ and its complement are maximum cliques.

2 Matrices over $\mathbb{Z}_{p^{s}}$

In this section, we recall some basic properties of matrices over $\mathbb{Z}_{p^{s}}$.

Lemma 2.1

(see [1, Proposition 6.2.2] or [20, p.328])*  Every non-zero element $x$ in $\mathbb{Z}_{p^{s}}$ can be written as $x=up^{t}$ where $u$ is a unit and $0\leq t\leq s-1$. Moreover, the integer $t$ is unique and $u$ is unique modulo the ideal $(p^{s-t})$ of $\mathbb{Z}_{p^{s}}$.*

Let $T_{p}=\{0,1,\ldots,p-1\}\subseteq\mathbb{Z}_{p^{s}}$. For two distinct elements $a,b\in T_{p}$, we always have $a-b\in\mathbb{Z}_{p^{s}}^{*}$. Without loss of generality, we may assume that $T_{p}=\mathbb{Z}_{p}$ in our discussion.

Lemma 2.2

(cf. [20, p.328])*   Every non-zero element $x$ in $\mathbb{Z}_{p^{s}}$ can be written uniquely as*

[TABLE]

where $t_{i}\in T_{p}$, $i=0,1,\ldots,s-1$.

By Lemma 2.2, every matrix $X\in\mathbb{Z}_{p^{s}}^{m\times n}$ can be written uniquely as

[TABLE]

where $X_{i}\in T_{p}^{m\times n}$, $i=0,\ldots,s-1$.

Note that every matrix in $T_{p}^{m\times n}$ can be seen as a matrix in $\mathbb{Z}_{p}^{m\times n}$. We define the natural surjection

[TABLE]

by $\pi(X)=X_{0}$ for all $X\in\mathbb{Z}_{p^{s}}^{m\times n}$ of the form (2.1). Clearly, $\pi(A)=A$ if $A\in\mathbb{Z}_{p}^{m\times n}$. For $X,Y\in\mathbb{Z}_{p^{s}}^{m\times n}$ and $Q\in\mathbb{Z}_{p^{s}}^{n\times k}$, We have

[TABLE]

[TABLE]

[TABLE]

By Lemma 2.1, it is easy to prove the following result.

Lemma 2.3

(cf. [21, Chap. II] or [20, p.327])*  Let $R=\mathbb{Z}_{p^{s}}$ where $s\geq 2$, and let $A\in R^{m\times n}$ be a non-zero matrix. Then there are $P\in GL_{m}(R)$ and $Q\in GL_{n}(R)$ such that*

[TABLE]

where $1\leq k_{1}\leq\cdots\leq k_{t}\leq s-1$. Moreover, the parameters $(r,t,k_{1},\ldots,k_{t})$ are uniquely determined by $A$. In (2.6), $I_{r}$ or ${\rm diag}\left(p^{k_{1}},\ldots,p^{k_{t}}\right)$ may be absent.

Let $A\in R^{m\times n}$, and let $I_{k}(A)$ be the ideal in $R$ generated by all $k\times k$ minors of $A$, $k=1,\ldots,{\rm min}\{m,n\}$. Let ${\rm Ann}_{R}(I_{k}(A))=\left\{x\in R:xI_{k}(A)=0\right\}$ denote the annihilator of $I_{k}(A)$. The McCoy rank of $A$, denoted by ${\rm rk}(A)$, is the following integer:

[TABLE]

We have that ${\rm rk}(A)={\rm rk}(^{t}A)$; ${\rm rk}(A)={\rm rk}(PAQ)$ where $P$ and $Q$ are invertible matrices of the appropriate sizes; and ${\rm rk}(A)=0$ if and only if ${\rm Ann}_{R}(I_{1}(A))\neq(0)$ (cf. [2]).

Lemma 2.4

(see [17, Lemmas 2.4 and 2.7])*  Let $0\neq A\in\mathbb{Z}_{p^{s}}^{m\times n}$ ($s\geq 2$) be of the form (2.6). Then $r+t$ is the inner rank of $A$, and $r$ is the McCoy rank of $A$.*

Let $A\in\mathbb{Z}_{p^{s}}^{m\times n}$. By Lemma 2.4, ${\rm rk}(A)\leq\rho(A)$, and ${\rm rk}(A)=0$ if and only if $A\in J_{p^{s}}^{m\times n}$. By Lemmas 2.3 and 2.4, it is easy to see that (cf. [17])

[TABLE]

For $A\in R^{m\times n}$ and $B\in R^{n\times m}$, if $AB=I_{m}$, we call that $A$ has a right inverse and $B$ is a right inverse of $A$. Similarly, if $AB=I_{m}$, than $B$ has a left inverse and $A$ is a left inverse of $B$. Note that if $a\in\mathbb{Z}_{p^{s}}^{*}$ and $b\in J_{p^{s}}$, then $a\pm b\in\mathbb{Z}_{p^{s}}^{*}$. By Lemma 2.3, we have the following lemmas.

Lemma 2.5

(cf. [17])*  Let $A\in\mathbb{Z}_{p^{s}}^{m\times n}$ where $s\geq 2$ and $n\geq m$. Then ${\rm rk}(A)={\rm rk}(A\pm B)$ for all $B\in J_{p^{s}}^{m\times n}$. Moreover, $A$ has a right inverse if and only if ${\rm rk}(A)=m$.*

Lemma 2.6

(see. [17, Lemma 4.2])*   If $A\in GL_{n}(\mathbb{Z}_{p^{s-1}})$ where $s\geq 2$, then $A\in GL_{n}(\mathbb{Z}_{p^{s}})$.*

Lemma 2.7

(see [17, Lemma 4.3])*   If $A\in\mathbb{Z}_{p^{s-1}}^{m\times n}$ where $s\geq 2$, then both $A$ and $Ap$ can be viewed as matrices in $\mathbb{Z}_{p^{s}}^{m\times n}$ with the same inner rank.*

3 Independence number of $\Gamma_{d}$ and MDR codes

3.1 Independence number and chromatic number of $\Gamma_{d}$

Recall that an independent set of a graph $G$ is a subset of vertices such that no two vertices are adjacent. A largest independent set of $G$ is an independent set of maximum cardinality. The independence number of $G$, denoted by $\alpha(G)$, is the number of vertices in a largest independent set of $G$.

An $r$-colouring of a graph $G$ is a homomorphism from $G$ to the complete graph $K_{r}$. The chromatic number of $G$, denoted by $\chi(G)$, is the least value $k$ for which $G$ can be $k$-coloured.

A clique of a graph $G$ is a complete subgraph of $G$. A clique $C$ is maximal if there is no clique of $G$ which properly contains $C$ as a subset. A maximum clique of $G$ is a clique of $G$ which has maximum cardinality. The clique number of $G$, denoted by $\omega(G)$, is the number of vertices in a maximum clique. For convenience, we regard that a maximal clique and its vertex set are the same.

Theorem 3.1

Every generalized bilinear forms graph $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ is a connected vertex transitive graph.

*Proof. *Let $R=\mathbb{Z}_{p^{s}}$. For any vertex $A$ of $\Gamma_{d}$, since the map $X\mapsto X-A$ is an automorphism of $\Gamma_{d}$, $\Gamma_{d}$ is vertex-transitive.

Let $A,B\in R^{m\times n}$ with $\rho(A-B)=r>0$. By Lemma 2.3, there are $P\in GL_{m}(R)$ and $Q\in GL_{n}(R)$ such that $B-A=P{\rm diag}\left(p^{k_{1}},\ldots,p^{k_{r}},0,\ldots,0\right)Q$, where $0\leq k_{1}\leq\cdots\leq k_{r}\leq{\rm max}\{s-1,1\}$. If $r\leq d-1$, then $A\sim B$. Now we assume that $r>d-1$. Put $A_{0}=A$, $A_{i}=A+P{\rm diag}\left(p^{k_{1}},\ldots,p^{k_{i}},0,\ldots,0\right)Q$, $i=1,\ldots,r$. Then $A_{r}=B$ and $A_{i}\sim A_{i+1}$, $i=0,1,\ldots,r-1$. It follows that $\Gamma_{d}$ is connected. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

For a graph $G$, we have (see [4, Theorem 6.10, Corollary 6.2])

[TABLE]

From [23, Lemma 2.7.2] we have

[TABLE]

Now, we recall the coding theory on a finite field $\mathbb{F}_{q}$ (cf. [7, 12, 15, 22]). Without loss of generality we assume that $n\geq m\geq d>1$ are integers. An $(m\times n,d)$ rank distance code over $\mathbb{F}_{q}$ is a subset $\mathcal{C}$ of $\mathbb{F}_{q}^{m\times n}$ with ${\rm d_{R}}(A,B)\geq d$ for distinct $A,B\in\mathcal{C}$. For an $(m\times n,d)$ rank distance code $\mathcal{C}$ over $\mathbb{F}_{q}$, we have $|\mathcal{C}|\leq q^{n(m-d+1)}$, and the bound $q^{n(m-d+1)}$ is called the Singleton bound for $\mathcal{C}$. If a rank distance code $\mathcal{C}$ over $\mathbb{F}_{q}$ satisfies $|\mathcal{C}|=q^{n(m-d+1)}$, then $\mathcal{C}$ is called an $(m\times n,d)$ maximum rank distance code (MRD code in short) over $\mathbb{F}_{q}$. A linear code of length $n^{\prime}$ over $\mathbb{F}_{q}$ is a subspace of $\mathbb{F}_{q}^{n^{\prime}}$.

MRD codes (over $\mathbb{F}_{q}$) can be used to correct errors and erasures in network. In 1978, Delsarte [8] (and independently Gabidulin in 1985 [11], Roth in 1991 [24]) proved the following important result:

Lemma 3.2

There are linear $(m\times n,d)$ MRD codes over $\mathbb{F}_{q}$ for all choices of $m,n,d$.

These linear MRD codes in Lemma 3.2 are also called Gabidulin codes. Until a few years ago, the only known MRD codes were Gabidulin codes. Recently, the research of MRD codes is active, and we know about other some MRD codes over $\mathbb{F}_{q}$ (cf. [7, 15, 22]).

Clearly, every independent set of $\Gamma_{d}(\mathbb{F}_{q}^{m\times n})$ is an $(m\times n,d)$ rank distance code over $\mathbb{F}_{q}$ and vice versa. In other words, an $(m\times n,d)$ MRD code over $\mathbb{F}_{q}$ is a largest independent set of $\Gamma_{d}(\mathbb{F}_{q}^{m\times n})$. Thus, Lemma 3.2 implies that

[TABLE]

The formula (3.3) was also showed by [12].

By (3.2) and (3.3), we get $\omega\left(\Gamma_{d}(\mathbb{F}_{q}^{m\times n})\right)\leq q^{n(d-1)}$. On the other hand, \small\mathcal{M}:=\left(\begin{array}[]{c}\mathbb{F}_{q}^{(d-1)\times n}\\ 0\\ \end{array}\right) is a clique and $|\mathcal{M}|=q^{n(d-1)}$. Thus we obtain

[TABLE]

Theorem 3.3

When $n\geq m$, the independence number and the clique number of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ are

[TABLE]

and

[TABLE]

*Proof. *We prove (3.5) and (3.6) by induction on $s$. When $s=1$, (3.5) and (3.6) hold by (3.3) and (3.4). Suppose that $s\geq 2$ and

[TABLE]

Write $\alpha=p^{n(m-d+1)}$. Let $\mathcal{A}_{1}=\left\{S_{1},S_{2},\ldots,S_{\alpha}\right\}$ be a largest independent set of $\Gamma_{d}(\mathbb{Z}_{p}^{m\times n})$, where $S_{i}\in\mathbb{Z}_{p}^{m\times n}$, $i=1,\ldots,\alpha$. Since each $S_{i}$ can be seen as an element in $\mathbb{Z}_{p^{s}}^{m\times n}$, $\mathcal{A}_{1}$ is also an independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$.

Put $\beta=p^{(s-1)n(m-d+1)}$. Let $\mathcal{A}_{s-1}=\{C_{1},C_{2},\ldots,C_{\beta}\}$ be a largest independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$. Note that every matrix over $\mathbb{Z}_{p^{s-1}}$ can be seen as a matrix over $\mathbb{Z}_{p^{s}}$. For any two distinct matrices $C_{i},C_{j}\in\mathcal{A}_{s-1}$, we have $\rho(C_{i}-C_{j})\geq d$. From Lemma 2.7 we get $\rho(C_{i}p-C_{j}p)\geq d$. Thus $\mathcal{A}_{s-1}p:=\{Xp:X\in\mathcal{A}_{s-1}\}$ is an independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ with $|\mathcal{A}_{s-1}p|=\beta$. Let

[TABLE]

Then $\mathcal{L}_{k}$ is an independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ and $|\mathcal{L}_{k}|=\beta$, $k=1,\ldots,\alpha$. Let

[TABLE]

Since ${\rm rk}(S_{i}-S_{j})\geq d$ for $i\neq j$, Lemma 2.5 implies that $\mathcal{A}_{s}$ is an independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ with $|\mathcal{A}_{s}|=\alpha\beta=p^{sn(m-d+1)}$. Therefore,

[TABLE]

By (3.2) and (3.9), we have

[TABLE]

On the other hand, it is easy to see that \small\mathcal{M}:=\left(\begin{array}[]{c}\mathbb{Z}_{p^{s}}^{(d-1)\times n}\\ 0\\ \end{array}\right) is a clique of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ and $|\mathcal{M}|=p^{sn(d-1)}$. Thus we obtain

[TABLE]

Using (3.2) and (3.10), we get

[TABLE]

It follows from (3.9) that (3.5) holds. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Let $G$ be a finite group and let $C$ be a subset of $G$ that is closed under taking inverses and does not contain the identity. The Cayley graph $X(G,C)$ is the graph with vertex set $G$ and two vertices $h$ and $g$ are adjacent if $hg^{-1}\in C$. A Cayley graph $X(G,C)$ is normal if $gCg^{-1}=C$ for all $g\in G$. It is easy to see that $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ is a normal Cayley graph on the matrix additive group $G$ of $\mathbb{Z}_{p^{s}}^{m\times n}$ and the inverse closed subset $C$ is the set of all matrices of inner rank $<d$.

Lemma 3.4

(Godsil [13, Corollary 6.1.3], cf. [23, Therem 3.3.1])*  Let $G$ be a normal Cayley graph. If $\alpha(G)\omega(G)=|V(G)|$, then $\chi(G)=\omega(G)$.*

By Theorem 3.3 and Lemma 3.4, the chromatic number of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ is

[TABLE]

3.2 MDR codes over $\mathbb{Z}_{p^{s}}$

We recall the usual definition of MDS code in coding theory (cf. [25]). An $(n,M)$ code over a finite alphabet $F$ is a nonempty subset $\mathcal{C}$ of size $M$ of $F^{n}$. For codewords $\alpha=(a_{1},\ldots,a_{n})$, $\beta=(b_{1},\ldots,b_{n})$, the Hamming distance between $\alpha$ and $\beta$ is $d(\alpha,\beta)=\left|\{i:a_{i}\neq b_{i},1\leq i\leq n\}\right|$. An $(n,M)$ code with minimum distance $d$ is called an $(n,M,d)$* code*. For any $(n,M,d)$ code over an alphabet of size $q$, we have the Singleton bound

[TABLE]

An $(n,M,d)$ code over an alphabet of size $q$ is called a maximum distance separable code (MDS code in short) if it attains the Singleton bound, i.e. $d=n-({\rm log}_{q}M)+1$.

A code $\mathcal{C}$ (over $\mathbb{Z}_{p^{s}}$) of length $m$ is a subset of $\mathbb{Z}_{p^{s}}^{m\times 1}$ (or $\mathbb{Z}_{p^{s}}^{m}$). If the code is a submodule (i.e. vector space) over $\mathbb{Z}_{p^{s}}$ we say that it is a linear code over $\mathbb{Z}_{p^{s}}$. Suppose that $n\geq m\geq d>1$ are integers. As a natural extension of rank distance code over $\mathbb{F}_{q}$, we define the rank distance code over $\mathbb{Z}_{p^{s}}$ as follows. An $(m\times n,d)$ rank distance code over $\mathbb{Z}_{p^{s}}$ is a subset $\mathcal{C}$ of $\mathbb{Z}_{p^{s}}^{m\times n}$ with ${\rm d_{R}}(A,B)\geq d$ for distinct $A,B\in\mathcal{C}$.

Every independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ is an $(m\times n,d)$ rank distance code over $\mathbb{Z}_{p^{s}}$ and vice versa. Thus, for an $(m\times n,d)$ rank distance code $\mathcal{C}$ over $\mathbb{Z}_{p^{s}}$, we have $|\mathcal{C}|\leq p^{sn(m-d+1)}$ by (3.5). If a rank distance code $\mathcal{C}$ over $\mathbb{Z}_{p^{s}}$ satisfies $|\mathcal{C}|=q^{n(m-d+1)}$, then $\mathcal{C}$ is called an $(m\times n,d)$ maximum rank distance code (MRD code in short) over $\mathbb{Z}_{p^{s}}$. In other words, an $(m\times n,d)$ MRD code over $\mathbb{Z}_{p^{s}}$ is a largest independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$.

Theorem 3.5

If $\mathcal{S}$ is a largest independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$, then $\mathcal{S}$ is both an $(m\times n,d)$ MRD code over $\mathbb{Z}_{p^{s}}$ and an $(m,M,d)$ MDS code with $M=p^{sn(m-d+1)}$. Moreover, there is a largest independent set $\mathcal{C}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ such that $\mathcal{C}$ is a linear code over $\mathbb{Z}_{p^{s}}$.

*Proof. *Step 1.  Without loss of generality we assume that $n\geq m\geq d>1$ are integers. By (3.5), it is clear that every largest independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ is an $(m\times n,d)$ MRD code over $\mathbb{Z}_{p^{s}}$. We prove that every largest independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ is a usual MDS code as follows.

By Lemma 2.2 and (2.1), the row vector space $\mathbb{Z}_{p^{s}}^{n}$ is isomorphic to $\mathbb{Z}_{p^{sn}}$ (as an $n$-dimensional vector space over $\mathbb{Z}_{p^{s}}$) (cf. [27, Chapter 14]). Thus, it is easy to see that $\mathbb{Z}_{p^{s}}^{m\times n}$ (as a vector space over $\mathbb{Z}_{p^{s}}$) is isomorphic to the column vector space $\mathbb{Z}_{p^{sn}}^{m\times 1}$ (as a vector space over $\mathbb{Z}_{p^{s}}$). Let $\mathcal{S}$ be a largest independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$. From (3.5) we have $|\mathcal{S}|=p^{sn(m-d+1)}$. For two distinct vertices $A,B\in\mathcal{S}$, since $\rho(A-B)\geq d$, it is clear that $A-B$ has at least $d$ non-zero rows. Hence the Hamming distance between $A$ and $B$ (as vectors in $\mathbb{Z}_{p^{sn}}^{m\times 1}$) is at least $d$. Thus, $\mathcal{S}$ can be seen as an $(m,M,d)$ code over an alphabet of size $p^{sn}$, where $M=|\mathcal{S}|=p^{sn(m-d+1)}$. Since $m-({\rm log}_{p^{sn}}M)+1=d$, $\mathcal{S}$ is a usual MDS code.

Step 2.  We assert that there is a largest independent set $\mathcal{C}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ such that it is a linear codes over $\mathbb{Z}_{p^{s}}$. By Lemma 3.2, we may assume with no loss of generality that $s\geq 2$. Using Lemma 3.2, there is a linear $(m\times n,d)$ MRD code $\mathcal{A}_{1}$ over $\mathbb{Z}_{p}$. Clearly, $\mathcal{A}_{1}$ is a vector space over $\mathbb{Z}_{p}$ and a largest independent set of $\Gamma_{d}(\mathbb{Z}_{p}^{m\times n})$.

When $s=2$, recalling (3.8), $\mathcal{A}_{2}:=\mathcal{A}_{1}+\mathcal{A}_{1}p$ is a largest independent set of $\Gamma_{d}(\mathbb{Z}_{p^{2}}^{m\times n})$. Since $\mathcal{A}_{1}$ contains [math], we have $\mathcal{A}_{1}\subset\mathcal{A}_{2}$. By Lemma 2.2 and (2.1), one can prove that $\mathcal{A}_{2}$ is a vector space over $\mathbb{Z}_{p^{2}}$. For $s-1$ ($s\geq 3$), assume that there is a largest independent set $\mathcal{A}_{s-1}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$, such that $\mathcal{A}_{1}\subset\mathcal{A}_{s-1}$ and $\mathcal{A}_{s-1}$ is a vector space over $\mathbb{Z}_{p^{s-1}}$. Then by (3.8), $\mathcal{A}_{s}:=\mathcal{A}_{1}+\mathcal{A}_{s-1}p$ is a largest independent set of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$. Since $\mathcal{A}_{s-1}$ contains [math], $\mathcal{A}_{1}\subset\mathcal{A}_{s}$. Applying Lemma 2.2 and (2.1), it is easy to prove that $\mathcal{A}_{s}$ is a vector space over $\mathbb{Z}_{p^{s}}$. By the induction on $s$, for any $s\geq 2$, there exists a largest independent set $\mathcal{A}_{s}:=\mathcal{A}_{1}+\mathcal{A}_{s-1}p$ of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$, such that $\mathcal{A}_{s}$ is a vector space over $\mathbb{Z}_{p^{s}}$. By Step 1, $\mathcal{A}_{s}$ is a linear code over $\mathbb{Z}_{p^{s}}$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

By Theorem 3.5, there are linear $(m\times n,d)$ MRD codes over $\mathbb{Z}_{p^{s}}$ for all choices of $m,n,d$. Applying (3.8), we can Construct many MRD codes over $\mathbb{Z}_{p^{s}}$.

4 Maximum cliques and Core of $\Gamma_{d}$

4.1 Maximum cliques of $\Gamma_{d}$

Note that the algebraic structures of maximum cliques of $\Gamma_{d}$ have many applications. For example, a maximum clique of $\Gamma_{d}$ is a largest independent set of the complement of $\Gamma_{d}$. We will determine the algebraic structures of maximum cliques of $\Gamma_{d}$.

Let $X=[\alpha_{1},\ldots,\alpha_{m}]$ be an $m$-dimensional vector subspace of $\mathbb{F}_{q}^{n}$ $(n\geq m)$. Then $X$ has a matrix representation \scriptsize\left(\begin{array}[]{c}\alpha_{1}\\ \vdots\\ \alpha_{m}\\ \end{array}\right)\in\mathbb{F}_{q}^{m\times n}. For simpleness, the matrix representation of a subspace $X$ is also denoted by $X$. If $X$ is a matrix representation of a subspace $X$ of $\mathbb{F}_{q}^{n}$, then $PX$ is also a matrix representation of $X$ where $P\in GL_{m}(\mathbb{F}_{q})$. It follows that the matrix representation is not unique. However, a subspace $X$ of $\mathbb{F}_{q}^{n}$ has a unique matrix representation which is the row-reduced echelon form $X=(I_{m},B)Q$, where $Q$ is a permutation matrix.

A geometric description of the bilinear forms graph $\Gamma_{2}(\mathbb{F}_{q}^{m\times n})$ is the adjacency graph of the attenuated space. Let $n\geq m$ and let

[TABLE]

be a fixed $n$-dimension subspace of $\mathbb{F}_{q}^{m+n}$. Write

[TABLE]

The incidence structure $\left(\mathcal{A}_{m},\mathcal{A}_{m-1},\subseteq\right)$ is called an attenuated space. Its adjacency graph is the graph with $\mathcal{A}_{m}$ as its vertex set, and two vertices being adjacent if their intersection is in $\mathcal{A}_{m-1}$. For any $U\in\mathcal{A}_{m}$, by $U\cap W=\{0\}$ we have $U=(X_{U},I_{m})$, where $X_{U}\in\mathbb{F}_{q}^{m\times n}$ is uniquely determined by $U$. Let

[TABLE]

Then $\varphi$ is a graph isomorphism from the adjacency graph of $\left(\mathcal{A}_{m-1},\mathcal{A}_{m},\subseteq\right)$ to $\Gamma_{2}(\mathbb{F}_{q}^{m\times n})$. Moreover, ${\rm dim}(U_{1}\cap U_{2})=r$ if and only if ${\rm rank}(\varphi(U_{1})-\varphi(U_{2}))=m-r$ for all $U_{1},U_{2}\in\mathcal{A}_{m}$ (cf. [18], [3, §9.5A]).

Lemma 4.1

(see [26, Theorem 3(2)])*  Suppose that $n\geq m$ and $\Gamma$ is the adjacency graph of an attenuated space $\left(\mathcal{A}_{m},\mathcal{A}_{m-1},\subseteq\right)$. Let $\mathscr{F}$ be a collection of elements of the $\mathcal{A}_{m}$ with the property that ${\rm dim}(\gamma\cap\delta)\geq t$ for all $\gamma,\delta$ in $\mathscr{F}$. Then $|\mathscr{F}|\leq q^{(m-t)n}$, and equality holds if and only if either (a) $\mathscr{F}$ consists of all elements of $\mathcal{A}_{m}$ which contain a fixed $t$-dimensional subspace $U$ with $U\cap W=0$, or (b) $m=n$ and $\mathscr{F}$ is the set of all elements of $\mathcal{A}_{m}$ contained in a fixed $(2m-t)$-dimensional subspace $U^{\prime}$ with ${\rm dim}(U^{\prime}\cap W)=m-t$.*

Lemma 4.2

In $\Gamma_{d}(\mathbb{F}_{q}^{m\times n})$ (where $n\geq m$), every maximum clique ${\cal M}$ containing [math] is of the form either

[TABLE]

where $P\in GL_{m}(\mathbb{F}_{q})$ is fixed; or

[TABLE]

with $n=m$, where $Q\in GL_{m}(\mathbb{F}_{q})$ is fixed.

*Proof. *Let $\varphi$ be the graph isomorphism (4.1) from the adjacency graph of $\left(\mathcal{A}_{m-1},\mathcal{A}_{m},\subseteq\right)$ to the bilinear forms graph $\Gamma_{2}(\mathbb{F}_{q}^{m\times n})$. For any $A,B\in\mathbb{F}_{q}^{m\times n}$, we have ${\rm rank}(A-B)<d$ (where $2\leq d\leq m$) if and only if ${\rm dim}(\varphi^{-1}(A)\cap\varphi^{-1}(B))\geq m-d+1$. Thus, ${\cal M}$ is a maximum clique of $\Gamma_{d}(\mathbb{F}_{q}^{m\times n})$ if and only if $\varphi^{-1}({\cal M})$ has the property that ${\rm dim}(\gamma\cap\delta)\geq m-d+1$ for all $\gamma,\delta$ in $\varphi^{-1}({\cal M})$ and $|\varphi^{-1}({\cal M})|={\rm max}\{|\mathscr{F}|\}$ for all collection $\mathscr{F}$ of elements of $\mathcal{A}_{m}$ with the property that ${\rm dim}(\gamma\cap\delta)\geq m-d+1$ for all $\gamma,\delta$ in $\mathscr{F}$.

Let ${\cal M}$ be a maximum clique containing [math] in $\Gamma_{d}(\mathbb{F}_{q}^{m\times n})$. Write $W=(I_{n},0)$. By Lemma 4.1 and above result, $\varphi^{-1}({\cal M})$ is of the form either (a) $\varphi^{-1}({\cal M})$ consists of all elements of $\mathcal{A}_{m}$ which contain a fixed $(m-d+1)$-dimensional subspace $U$ with $U\cap W=0$, or (b) $m=n$ and $\varphi^{-1}({\cal M})$ is the set of all elements of $\mathcal{A}_{m}$ contained in a fixed $(m+d-1)$-dimensional subspace $U^{\prime}$ with ${\rm dim}(U^{\prime}\cap W)=d-1$. In the case (a), it is easy to see that ${\cal M}$ is of the form (4.2). Now, we assume the case (b) happens. Since ${\rm dim}(U^{\prime}\cap W)=d-1$, by appropriate elementary operations of matrix, we may assume with no loss of generality that \small U^{\prime}=\left(\begin{array}[]{ccc}I_{d-1}&0&0\\ 0&0&I_{m}\\ \end{array}\right). Then $\varphi^{-1}({\cal M})=\left\{(Y,0,I_{m}):Y\in\mathbb{F}_{q}^{m\times(d-1)}\right\}$. Thus (4.3) holds. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

However, the proof of Lemma 4.2 cannot be generalized to the case of $\mathbb{Z}_{p^{s}}$. In order to generalize Lemma 4.2 to the case of $\mathbb{Z}_{p^{s}}$, we need a new method.

Lemma 4.3

Let $k\geq 2$ and $A\in\mathbb{Z}_{p}^{k\times k}$. Then there exists $B\in GL_{k}(\mathbb{Z}_{p})$ such that $A-B$ is invertible.

*Proof. *Without losing generality, we may assume that $A=I_{k}$. When $k$ is an even number, we have \small I_{k}-\left(\begin{array}[]{cc}I_{k/2}&I_{k/2}\\ I_{k/2}&0\\ \end{array}\right) is invertible, and hence this lemma holds. From now on we assume that $k$ is an odd number. When $k=3$, there is \scriptsize B_{1}:=\left(\begin{array}[]{ccc}1&1&0\\ 1&0&1\\ 0&1&0\\ \end{array}\right)\in GL_{3}(\mathbb{Z}_{p}) such that $I_{3}-B_{1}$ is invertible. When $k=2r+1$ ($r\geq 2$), by the case of even, there is $B_{2}\in GL_{2r-2}(\mathbb{Z}_{p})$ such that $I_{2r-2}-B_{1}$ is invertible, and hence $I_{2r+1}-{\rm diag}(B_{1},B_{2})$ is invertible. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Theorem 4.4

Let $n\geq m$, and let ${\cal M}$ be a maximum clique of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$. Then ${\cal M}$ is of the form either

[TABLE]

where $P\in GL_{m}(\mathbb{Z}_{p^{s}})$ and $B\in\mathbb{Z}_{p^{s}}^{m\times n}$ are fixed; or

[TABLE]

with $n=m$, where $Q\in GL_{m}(\mathbb{Z}_{p^{s}})$ and $B\in\mathbb{Z}_{p^{s}}^{m\times m}$ are fixed.

*Proof. *When $s=1$, this theorem holds by Lemma 4.2. By [17, Theorem 3.6], this theorem holds if $d=2$. Thus, from now on we assume that $s\geq 2$ and $m\geq d\geq 3$. We prove this theorem by induction on $s$. Assume that this theorem holds for $s-1$. We prove that it holds for $s$ as follows.

Let ${\cal M}=\{A_{1},\ldots,A_{\omega}\}$ be a maximum clique of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$. By (3.6) we have $\omega=|{\cal M}|=p^{sn(d-1)}$. By the bijection $X\mapsto X-B$, we can assume that ${\cal M}$ contains [math] (i.e., $B=0$).

Let $\pi:\mathbb{Z}_{p^{s}}^{m\times n}\rightarrow\mathbb{Z}_{p}^{m\times n}$ be the natural surjection (2.2). For any vertex $X$ in $\pi(\mathcal{M})$, let $\pi^{-1}(X)$ denote the preimages of $X$ in $\mathcal{M}$, i.e., $\pi^{-1}(X)=\left\{Y\in\mathcal{M}:\pi(Y)=X\right\}$. Suppose $\{\pi(A_{i_{1}}),\ldots,\pi(A_{i_{h}})\}$ is the set of all different elements in $\{\pi(A_{1}),\ldots,\pi(A_{\omega})\}$. Then $\pi({\cal M})=\{\pi(A_{i_{1}}),\ldots,\pi(A_{i_{h}})\}$ is a clique of $\Gamma_{d}(\mathbb{Z}_{p}^{m\times n})$. It follows from (3.4) that

[TABLE]

Clearly, ${\cal M}$ has a partition into $h$ cliques

[TABLE]

where $\pi({\cal M}_{t})=\{\pi(A_{i_{t}})\}$, $t=1,\ldots,h$. Thus

[TABLE]

Let $n_{t}=|{\cal M}_{t}|$ and let

[TABLE]

where $\left\{B_{1t},B_{2t},\ldots,B_{n_{t},t}\right\}$ is a clique in $J_{p^{s}}^{m\times n}$, $t=1,\ldots,h$. Write $B_{jt}=pC_{jt}$ where $C_{jt}\in\mathbb{Z}_{p^{s-1}}^{m\times n}$, $j=1,\ldots,n_{t}$, $t=1,\ldots,h$. By Lemma 2.7, $\left\{C_{1t},C_{2t},\ldots,C_{n_{t},t}\right\}$ is a clique of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$, $t=1,\ldots,h$. By (3.6), we get

[TABLE]

Thus $p^{sn(d-1)}=|{\cal M}|\leq hp^{(s-1)n(d-1)}$, and hence $p^{n(d-1)}\leq h$. By (4.6), we obtain

[TABLE]

Therefore, $\pi({\cal M})$ is a maximum clique containing [math] in $\Gamma_{d}(\mathbb{Z}_{p}^{m\times n})$.

By Lemma 4.2, we have either \small\pi({\cal M})=P_{0}\left(\begin{array}[]{c}\mathbb{Z}_{p}^{(d-1)\times n}\\ 0\\ \end{array}\right), or $\pi({\cal M})=\left(\mathbb{Z}_{p}^{m\times(d-1)},0\right)Q_{0}$ with $n=m$, where $P_{0},Q_{0}$ are invertible matrices over $\mathbb{Z}_{p}$. Thus, ${\cal M}$ contains a matrix $E$ of inner rank $d-1$, such that either $\pi(E)=P_{0}{\rm diag}(I_{d-1},0)$ or $\pi(E)={\rm diag}(I_{d-1},0)Q_{0}$. Using appropriate elementary operations of matrix, without losing generality, we may assume that

[TABLE]

Since \scriptsize E=P_{0}\left(\begin{array}[]{c}X_{1}\\ 0\\ \end{array}\right) where $X_{1}\in\mathbb{Z}_{p}^{(d-1)\times n}$, or $E=\left(Y_{1},0\right)Q_{0}$ with $n=m$, where $Y_{1}\in\mathbb{Z}_{p}^{m\times(d-1)}$, it follows that \scriptsize P_{0}=\left(\begin{array}[]{cc}P_{11}&*\\ 0&*\\ \end{array}\right) or \scriptsize Q_{0}=\left(\begin{array}[]{cc}Q_{11}&0\\ &*\\ \end{array}\right), where $P_{11},Q_{11}\in GL_{d-1}(\mathbb{Z}_{p})$. Thus, $\pi({\cal M})$ is of the form either

[TABLE]

or

[TABLE]

By (4.7)–(4.9), it is easy to see that

[TABLE]

Thus

[TABLE]

It follows that $\left\{C_{1t},C_{2t},\ldots,C_{n_{t},t}\right\}$ is a maximum clique of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$, $t=1,\ldots,h$. In other words, there exists a maximum clique $\mathcal{C}_{t}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$ such that

[TABLE]

By the induction hypothesis, $\mathcal{C}_{t}$ is of the form either (4.4), or (4.5) with $n=m$. Thus,

[TABLE]

where $P_{t1},Q_{t1}\in\mathbb{Z}_{p^{s-1}}^{(d-1)\times(d-1)}$, \scriptsize\left(\begin{array}[]{c}P_{t1}\\ P_{t2}\\ \end{array}\right)\in\mathbb{Z}_{p^{s-1}}^{m\times(d-1)} has a left inverse, $(Q_{t1},Q_{t2})\in\mathbb{Z}_{p^{s-1}}^{(d-1)\times n}$ has a right inverse, and $B_{t}\in\mathbb{Z}_{p^{s-1}}^{m\times n}$, $t=1,\ldots,h$.

Let $A_{i_{1}}=0$. Then $\pi(0)=0$, $\pi^{-1}(0)$ is of the form (4.14) and $\mathcal{C}_{1}$ contains [math]. By (4.15), we have either

[TABLE]

or

[TABLE]

Suppose (4.16) holds. Since (4.10) and \small\left(\begin{array}[]{cc}0&P_{11}Yp\\ 0&P_{12}Yp\\ \end{array}\right)\sim\left(\begin{array}[]{cc}I_{d-1}&0\\ 0&0\\ \end{array}\right), \small\rho\left(\begin{array}[]{cc}-I_{d-1}&P_{11}Yp\\ 0&P_{12}Yp\\ \end{array}\right)\leq d-1 for all $Y\in\mathbb{Z}_{p^{s-1}}^{(d-1)\times(n-d+1)}$. By (2.7), we get that $P_{12}=0$ and $P_{11}$ is invertible. Therefore

[TABLE]

Suppose (4.17) holds. We have similarly that

[TABLE]

Thus, $\pi^{-1}(0)$ is of the form either (4.18) or (4.19) with $n=m$.

We distinguish the following two cases to prove this theorem.

Case 1.   $\pi({\cal M})$ is of the form (4.11).

First, we show that (4.18) holds. Let \scriptsize D=\left(\begin{array}[]{cc}0&I_{d-1}\\ 0&0\\ \end{array}\right)\in\pi({\cal M}). By (4.14), there is a maximum clique $\mathcal{C}_{D}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$ such that

[TABLE]

Suppose that $\pi^{-1}(0)$ is of the form (4.19) with $n=m$. Then \small\left(\begin{array}[]{cc}X_{1}p&I_{d-1}+X_{2}p\\ X_{3}p&X_{4}p\\ \end{array}\right)\sim\left(\begin{array}[]{cc}Xp&0\\ Wp&0\\ \end{array}\right) for all $X\in\mathbb{Z}_{p^{s-1}}^{(d-1)\times(d-1)}$ and $W\in\mathbb{Z}_{p^{s-1}}^{(m-d+1)\times(d-1)}$. Thus, we can choose $X$ and $W$ such that

[TABLE]

where $0\neq W_{1}\in\mathbb{Z}_{p^{s-1}}^{(m-d+1)\times 1}$. By (2.7), this is a contradiction. Thus $\pi^{-1}(0)$ must be of the form (4.18).

Let $e_{i}$ be the $i$-th column of $I_{d-1}$. By (4.14) and (4.15), there is a maximum clique $\mathcal{C}_{e_{i}}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$ such that

[TABLE]

Let $T$ be any $(n-1)\times(n-1)$ permutation matrix over $\mathbb{Z}_{p}$. Then $\rho\left({\rm diag}(0,I_{d-2})Tp,\ e_{1}\right)=d-1$. Write $X_{2}=X_{2}^{\prime}T$, $X_{2}\in\mathbb{Z}_{p^{s-1}}^{(m-d+1)\times(n-1)}$. Using (4.18), we get

[TABLE]

Thus

[TABLE]

where $X_{21}^{\prime}\in\mathbb{Z}_{p^{s-1}}^{(m-d+1)\times(n-d+1)}$. It follows from (2.7) that $X_{21}^{\prime}=0$, and hence the matrix $X_{2}$ has $n-d+1$ columns to be zeros. Since the permutation matrix $T$ is arbitrary, every column of $X_{2}$ must be zero, and hence $X_{2}=0$. Then

[TABLE]

Since \small\left(\begin{array}[]{cc}X_{1}p&e_{1}+Y_{3}p\\ 0&Y_{4}p\\ \end{array}\right)\sim\left(\begin{array}[]{cc}I_{d-1}&0\\ 0&0\\ \end{array}\right), it is easy to see that $Y_{4}p=0$. Therefore, we obtain that

[TABLE]

Similarly, we can prove that

[TABLE]

Let $A\in GL_{d-1}(\mathbb{Z}_{p})$. By (4.14), there is a maximum clique $\mathcal{C}_{A}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$ such that

[TABLE]

By (4.18), we have \bordermatrix{&{}_{d-1}&{}_{n-d}&{}_{1}\cr&A+X_{1}p&Y_{1}p&Z_{1}p\cr&X_{2}p&Y_{2}p&Z_{2}p\cr}\ \sim\bordermatrix{&{}_{d-1}&{}_{n-d}&{}_{1}\cr&X_{1}p&Y_{1}p&Z_{1}p\cr&0&0&0\cr}. Thus \small\rho\left(\begin{array}[]{ccc}A&0&0\\ X_{2}p&Y_{2}p&Z_{2}p\\ \end{array}\right)\leq d-1, and hence $Y_{2}p=0$ and $Z_{2}p=0$. On the other hand, from (4.21) we get

[TABLE]

Consequently \small\rho\left(\begin{array}[]{ccc}A&0&-e_{i}\\ X_{2}p&0&0\\ \end{array}\right)=\rho\left(\begin{array}[]{ccc}A&0&-e_{i}\\ 0&0&X_{2}A^{-1}e_{i}p\\ \end{array}\right)\leq d-1, and hence $X_{2}A^{-1}e_{i}=0$ by (2.7), $i=1,\ldots,d-1$. Therefore $X_{2}=0$. Then we obtain

[TABLE]

Let $A\in GL_{d-1}(\mathbb{Z}_{p})$. Write $F_{i}=(A,e_{i})$. By (4.14), there is a maximum clique $\mathcal{C}_{F_{i}}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$ such that

[TABLE]

From (4.21) we have \bordermatrix{&{}_{d-1}&{}_{n-d}&{}_{1}\cr&A+X_{1}p&Y_{1}p&e_{i}+Z_{1}p\cr&X_{2}p&Y_{2}p&Z_{2}p\cr}\ \sim\bordermatrix{&{}_{d-1}&{}_{n-d}&{}_{1}\cr&X_{1}p&Y_{1}p&e_{i}+Z_{1}p\cr&0&0&0\cr}, which implies that $Y_{2}p=0$ and $Z_{2}p=0$. By (4.18), we get \bordermatrix{&{}_{d-1}&{}_{n-d}&{}_{1}\cr&A+X_{1}p&Y_{1}p&e_{i}+Z_{1}p\cr&X_{2}p&0&0\cr}\ \sim\bordermatrix{&{}_{d-1}&{}_{n-d}&{}_{1}\cr&X_{1}p&Y_{1}p&Z_{1}p\cr&0&0&0\cr}, and thus

[TABLE]

By (2.7), one has $X_{2}A^{-1}e_{i}=0$. Using (4.21) again, we get

[TABLE]

and hence

[TABLE]

It follows from $X_{2}A^{-1}e_{i}=0$ that $X_{2}A^{-1}e_{j}=0$, $j=1,\ldots,d-1$, which implies that $X_{2}=0$. Thus, for any $A\in GL_{d-1}(\mathbb{Z}_{p})$, we have

[TABLE]

Let $B\in\mathbb{Z}_{p}^{(d-1)\times(d-1)}$. By (4.14), there is a maximum clique $\mathcal{C}_{B}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$ such that

[TABLE]

By Lemma 4.3, exists $A\in GL_{d-1}(\mathbb{Z}_{p})$ such that $B-A$ is invertible. Applying (4.22), we get

[TABLE]

which implies that $Y_{2}p=0$ and $Z_{2}p=0$. On the other hand, from (4.23) we have

[TABLE]

it follows that

[TABLE]

By (2.7), we obtain $X_{2}(B-A)^{-1}=0$ and hence $X_{2}=0$. Then we have proved that

[TABLE]

In particular, we have

[TABLE]

Using (4.25) and (4.18), similar to the proof of (4.22), we can get that

[TABLE]

Let $A\in GL_{d-1}(\mathbb{Z}_{p})$. Similar to the proof of (4.23), by (4.18) and (4.25) we can prove that

[TABLE]

Using (4.26) and (4.27), similar to the proof of (4.24), we have

[TABLE]

Now, let $A\in GL_{d-1}(\mathbb{Z}_{p})$ and $B_{1}\in\mathbb{Z}_{p}^{(d-1)\times(n-d+1)}$. Write $H=(A,B_{1})$. By (4.14), there is a maximum clique $\mathcal{C}_{H}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$ such that

[TABLE]

By (4.28), \left(\begin{array}[]{cc}A+X_{1}p&B_{1}+Y_{1}p\\ X_{2}p&Y_{2}p\\ \end{array}\right)\sim\left(\begin{array}[]{cc}X_{1}p&B_{1}+Y_{1}p\\ 0&0\\ \end{array}\right), hence \small\rho\left(\begin{array}[]{cc}A&0\\ X_{2}p&Y_{2}p\end{array}\right)=\rho\left(\begin{array}[]{cc}A&0\\ 0&Y_{2}p\end{array}\right)\leq d-1, which implies that $Y_{2}p=0$. By (4.28) again, \left(\begin{array}[]{cc}A+X_{1}p&B_{1}+Y_{1}p\\ X_{2}p&0\\ \end{array}\right)\sim\left(\begin{array}[]{cc}X_{1}p&B_{1}+(0,e_{i})+Y_{1}p\\ 0&0\\ \end{array}\right), and hence \small\rho\left(\begin{array}[]{ccc}A&0&-e_{i}\\ X_{2}p&0&0\end{array}\right)=\rho\left(\begin{array}[]{ccc}A&0&-e_{i}\\ 0&0&X_{2}A^{-1}e_{i}p\end{array}\right)\leq d-1, $i=1,\ldots,d-1$. It follows that $X_{2}A^{-1}e_{i}=0$, $i=1,\ldots,d-1$, thus $X_{2}=0$. Therefore, we get that

[TABLE]

for all $A\in GL_{d-1}(\mathbb{Z}_{p})$ and $B_{1}\in\mathbb{Z}_{p}^{(d-1)\times(n-d+1)}$.

Finally, let $B\in\mathbb{Z}_{p}^{(d-1)\times(d-1)}$ and $B_{1}\in\mathbb{Z}_{p}^{(d-1)\times(n-d+1)}$. Write $L=(B,B_{1})$. By (4.14), there is a maximum clique $\mathcal{C}_{L}$ of $\Gamma_{d}(\mathbb{Z}_{p^{s-1}}^{m\times n})$ such that

[TABLE]

By Lemma 4.3, there is $A\in GL_{d-1}(\mathbb{Z}_{p})$ such that $B-A$ is invertible. Applying (4.29),

[TABLE]

and hence \small\rho\left(\begin{array}[]{cc}B-A&0\\ X_{2}p&Y_{2}p\end{array}\right)=\rho\left(\begin{array}[]{cc}B-A&0\\ 0&Y_{2}p\end{array}\right)\leq d-1. Thus $Y_{2}p=0$. By (4.29) again, we have

[TABLE]

and hence

[TABLE]

Consequently, $X_{2}(B-A)^{-1}e_{i}p=0$, $i=1,\ldots,d-1$. Thus it is clear that $X_{2}=0$. Therefore, we obtain that

[TABLE]

for all $B\in\mathbb{Z}_{p}^{(d-1)\times(d-1)}$ and $B_{1}\in\mathbb{Z}_{p}^{(d-1)\times(n-d+1)}$. Then we have proved that

[TABLE]

Case 2.   $\pi({\cal M})$ is of the form (4.12) with $n=m$.

Let ${}^{t}{\cal M}=\{\,^{t}X:X\in{\cal M}\}$. Then ${}^{t}{\cal M}$ is a maximum clique of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times m})$. By (2.5), $\pi(^{t}{\cal M})=\,^{t}(\pi({\cal M}))$ is of the form (4.11). By Case 1, we have that {}^{t}{\cal M}=\left(\begin{array}[]{c}\mathbb{Z}_{p^{s}}^{(d-1)\times m}\\ 0\\ \end{array}\right). Thus ${\cal M}=\left(\mathbb{Z}_{p^{s}}^{(m)\times(d-1)},0\right)$. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

4.2 Cores of $\Gamma_{d}$ and the complement of $\Gamma_{d}$

A graph $G$ is a core [14] if every endomorphism of $G$ is an automorphism. A subgraph $H$ of a graph $G$ is a core of $G$ [14] if it is a core and there exists some homomorphism from $G$ to $H$. Every graph $G$ has a core, which is an induced subgraph and is unique up to isomorphism [14, Lemma 6.2.2].

Lemma 4.5

(see [23, Lemma 2.5.9])*  Let $G$ be a graph. Then the core of $G$ is the complete graph $K_{r}$ if and only if $\omega(G)=r=\chi(G)$.*

Corollary 4.6

Let $n\geq m$. Then the core of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ is a maximum clique of $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$, and $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ is not a core.

*Proof. *Let $\Gamma=\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$. By (3.11), we have $\chi(\Gamma)=\omega(\Gamma)$. By Lemma 4.5, the core of $\Gamma$ is a maximum clique, and hence $\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ is not a core. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$

Let $\overline{G}$ denote the complement of a graph $G$. Then $\alpha(\overline{G})=\omega(G)$, $\omega(\overline{G})=\alpha(G)$, and ${\rm Aut}(\overline{G})={\rm Aut}(G)$. Moreover, $\chi(\overline{G})\geq\omega(\overline{G})=\alpha(G)$.

Theorem 4.7

Let $\Gamma=\Gamma_{d}(\mathbb{Z}_{p^{s}}^{m\times n})$ where $n\geq m$. Then $\chi(\overline{\Gamma})=\alpha\left(\Gamma\right)=\omega(\overline{\Gamma})$. Moreover, the core of $\overline{\Gamma}$ is a maximum clique of $\overline{\Gamma}$, and $\overline{\Gamma}$ is not a core.

*Proof. *Let $\mathcal{S}=\left\{S_{1},S_{2},\ldots,S_{\alpha}\right\}$ be a largest independent set of $\Gamma$, where $\alpha:=\alpha(\Gamma)=p^{sn(m-d+1)}$. Let

[TABLE]

By (3.6), $\mathcal{M}_{1}$ is a maximum clique of $\Gamma$. Put $\mathcal{C}_{i}=\mathcal{M}_{1}+S_{i}$, $i=1,\ldots,\alpha$. Then $\mathcal{C}_{1},\ldots,\mathcal{C}_{\alpha}$ are $\alpha$ maximum cliques of $\Gamma$, and $\mathcal{C}_{i}\cap\mathcal{C}_{j}=\emptyset$ for all $i\neq j$. By Theorem 3.3, we get $\left|V(\Gamma)\right|=p^{smn}=\alpha\cdot\omega(\Gamma)$. Thus, $V(\Gamma)$ has a partition into $\alpha$ maximum cliques: $V(\Gamma)=\mathcal{C}_{1}\cup\mathcal{C}_{2}\cup\cdots\cup\mathcal{C}_{\alpha}$. Since $\mathcal{C}_{1},\ldots,\mathcal{C}_{\alpha}$ are $\alpha$ largest independent sets of $\overline{\Gamma}$, $V(\overline{\Gamma})$ has a partition into $\alpha$ largest independent sets: $V(\overline{\Gamma})=\mathcal{C}_{1}\cup\mathcal{C}_{2}\cup\cdots\cup\mathcal{C}_{\alpha}$. Hence $\chi(\overline{\Gamma})\leq\alpha$. By $\chi(\overline{\Gamma})\geq\alpha=\omega(\overline{\Gamma})$, we obtain $\chi(\overline{\Gamma})=\alpha\left(\Gamma\right)=\omega(\overline{\Gamma})$. By Lemma 4.5, we have similarly that the core of $\overline{\Gamma}$ is a maximum clique, and hence $\overline{\Gamma}$ is not a core. $\hfill\Box\vskip 6.0pt plus 2.0pt minus 2.0pt$