arXiv subjects

S. Morteza Mirafzal

Publications and source records attributed to S. Morteza Mirafzal.

At least 19 recordsLinked to original sources

arXiv2026

The distance spectrum of the bipartite double cover of strongly regular graphs

A strongly regular graph with parameters $(n,d,a,c)$ is a $d$-regular graph of order $n$, in which every pair of adjacent vertices has exactly $a$ common neighbor(s) and every pair of nonadjacent vertices has exactly $c$ common neighbor(s). Let $n$ be the number of vertices of the graph $G=(V,E)$. The distance matrix $D=D(G)$ of $G$ is an $n \times n $ matrix with the rows and columns indexed by $V$ such that $D_{uv} = d_{G}(u, v)=d(u,v)$, where $d_{G}(u, v)$ is the distance between the vertices $u$ and $v$ in the graph $G$. In this paper, we are interested in determining the distance spectrum of the bipartite double cover of the family of strongly regular graphs. In other words, let $G=(V,E)$ be a strongly regular graph with parameters $(n,k,a,c)$. We show that there is a close relationship between the spectrum of $G$ and the distance spectrum of $B(G)$, where $B(G)$ is the double cover of $G$. We explicitly determine the distance spectrum of the graph $B(G)$, according to the spectrum of $G$. In fact, according to the parameters of the graph $G$.

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arXiv2025

The distance spectrum of the line graph of the crown graph

The distance eigenvalues of a connected graph $G$ are the eigenvalues of its distance matrix $D(G)$. A graph is called distance integral if all of its distance eigenvalues are integers. Let $n \geq 3$ be an integer. The crown graph $Cr(n)$ is a graph obtained from the complete bipartite graph $K_{n,n}$ by removing a perfect matching. Let $L(Cr(n))$ denote the line graph of the crown graph $Cr(n)$. Using the equitable partition method, the set of distinct distance eigenvalues of the graph $L(Cr(n))$ has been determined which shows that this graph is distance integral [S.Morteza Mirafzal, The line graph of the crown graph is distance integral, Linear and Multilinear Algebra 71, no. 4 (2023): 662-672]. The distance spectrum of the graph $L(Cr(n))$ has not been found yet. In this paper, having the set of distance eigenvalues of $L(Cr(n))$ in the hand, we determine the distance spectrum of this graph.

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arXiv2024

Automorphism group of a family of distance regular graphs which are not distance transitive

Let $G_n=\mathbb{Z}_n\times \mathbb{Z}_n$ for $n\geq 4$ and $S=\{(i,0),(0,i),(i,i): 1\leq i \leq n-1\}\subset G_n$. Define $Γ(n)$ to be the Cayley graph of $G_n$ with respect to the connecting set $S$. It is known that $Γ(n)$ is a strongly regular graph with the parameters $(n^2,3n-3,n,6)$ \cite{19}. Hence $Γ(n)$ is a distance regular graph. It is known that every distance transitive graph is distance regular, but the converse is not true. In this paper, we study some algebraic properties of the graph $Γ(n)$. Then by determining the automorphism group of this family of graphs, we show that the graphs under study are not distance transitive.

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arXiv2022

Some remarks on the square graph of the hypercube

Let $Γ=(V,E)$ be a graph. The square graph $Γ^2$ of the graph $Γ$ is the graph with the vertex set $V(Γ^2)=V$ in which two vertices are adjacent if and only if their distance in $Γ$ is at most two. The square graph of the hypercube $Q_n$ has some interesting properties. For instance, it is highly symmetric and panconnected. In this paper, we investigate some algebraic properties of the graph ${Q^2_n}$. In particular, we show that the graph ${Q^2_n}$ is distance-transitive. We show that the graph ${Q^2_n}$ is an imprimitive distance-transitive graph if and only if $n$ is an odd integer. Also, we determine the spectrum of the graph $Q_n^2$. Finally, we show that when $n >2$ is an even integer, then ${Q^2_n}$ is an automorphic graph, that is, $Q_n^2$ is a distance-transitive primitive graph which is not a complete or a line graph.

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arXiv2022

$L(n)$ graphs are vertex-pancyclic and Hamilton-connected

A graph $G$ of order $n>2$ is pancyclic if $G$ contains a cycle of length $l$ for each integer $l$ with $3 \leq l \leq n $ and it is called vertex-pancyclic if every vertex is contained in a cycle of length $l$ for every $3 \leq l \leq n $. A graph $G$ of order $n > 2$ is Hamilton-connected if for any pair of distinct vertices $u$ and $v$, there is a Hamilton $u$-$v$ path, namely, there is a $u$-$v$ path of length $n-1$. The graph $ B(n)$ is a graph with the vertex set $V=\{v \ | \ v \subset [n] , | v | \in \{ 1,2 \} \} $ and the edge set $ E= \{ \{ v , w \} \ | \ v , w \in V , v \subset w $ or $ w \subset v \}$, where $[n]=\{1,2,...,n\}$. We denote by $L(n)$ the line graph of $B(n)$, that is, $L(n)=L(B(n))$. In this paper, we show that the graph $L(n)$ is vertex-pancyclic and Hamilton-connected whenever $n\geq 6$.

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arXiv2022

The automorphism group of the Andrásfai graph

Let $k \geq 1$ be an integer and $n=3k-1$. Let $\mathbb{Z}_n$ denote the additive group of integers modulo $n$ and let $C$ be the subset of $\mathbb{Z}_n$ consisting of the elements congruent to 1 modulo 3. The Cayley graph $Cay(\mathbb{Z}_n; C)$ is known as the Andr$\acute{a}$sfai graph And($k$). In this note, we determine the automorphism group of this graph. We will show that $Aut(And(k))$ is isomorphic with the dihedral group $\mathbb{D}_{2n}$.

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arXiv2022

On the distance eigenvalues of design graphs

A design graph is a regular bipartite graph in which any two distinct vertices of the same part have the same number of common neighbors. This class of graphs have a close relationship to strongly regular graphs. In this paper, we study the distance eigenvalues of the design graphs. Also, we will explicitly determine the distance eigenvalues of a class of design graphs, and determine the values for which the class is distance integral, that is, its distance eigenvalues are integers.

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arXiv2021

The line graph of the crown graph is distance integral

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arXiv2021

On The automorphism groups of $us$-Cayley graphs

Let $G$ be a finite abelian group written additively with identity $0$, and $Ω$ be an inverse closed generating subset of $G$ such that $0\notin Ω$. We say that $ Ω$ has the property \lq\lq{}$us$\rq\rq{} (unique summation), whenever for every $0 \neq g\in G$ if there are $s_1,s_2,s_3, s_4 \in Ω$ such that $s_1+s_2=g=s_3+s_4 $, then we have $\{s_1,s_2 \} = \{s_3,s_4 \}$. We say that a Cayley graph $Γ=Cay(G;Ω)$ is a $us$-$Cayley\ graph$, whenever $G$ is an abelian group and the generating subset $Ω$ has the property \lq\lq{}$us$\rq\rq{}. In this paper, we show that if $Γ=Cay(G;Ω)$ is a $us$-$Cayley\ graph$, then $Aut(Γ)=L(G)\rtimes A$, where $L(G)$ is the left regular representation of $G$ and $A$ is the group of all automorphism groups $θ$ of the group $G$ such that $θ(Ω)=Ω$. Then, as some applications, we explicitly determine the automorphism groups of some classes of graphs including Möbius ladders and $k$-ary $n$-cubes.

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arXiv2021

The spectrum of a class of graphs derived from Grassmann graphs

Let $n,k$ be positive integers such that $n\geq 3$, $k < \frac {n}{2} $. Let $q$ be a power of a prime $p$ and $\mathbb{F}_q$ be a finite field of order $q$. Let $V(q,n)$ be a vector space of dimension $n$ over $\mathbb{F}_q$. We define the graph $S(q,n,k)$ as a graph with the vertex set $V=V_k \cup V_{k+1}$, where $V_k$ and $V_{k+1}$ are the family of subspaces in $V(q,n)$ of dimension $k$ and $k+1$ respectively, in which two vertices $v$ and $w$ are adjacent whenever $v$ is a subspace of $w$ or $w$ is a subspace of $v$. It is clear that the graph $S(q,n,k)$ is a bipartite graph. In this paper, we study some properties of this graph. In particular, we determine the spectrum of the graph $S(q,n,k)$.

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arXiv2021

Some algebraic properties of a class of integral graphs determined by their spectrum

Let $Γ=(V,E)$ be a graph. If all the eigenvalues of the adjacency matrix of the graph $Γ$ are integers, then we say that $Γ$ is an integral graph. A graph $Γ$ is determined by its spectrum if every graph cospectral to it is in fact isomorphic to it. In this paper, we investigate some algebraic properties of the Cayley graph $Γ=Cay(\mathbb{Z}_{n}, S)$, where $n=p^m$, ($p$ is a prime integer, $m\in\mathbb{N}$) and $S=\{{a}\in\mathbb{Z}_{n}\,|\,\, (a, n)=1\}$. First, we show that $Γ$ is an integral graph. Also we determine the automorphism group of $Γ$. Moreover, we show that $Γ$ and $K_v \bigtriangledownΓ$ are determined by their spectrum.

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arXiv2020

On the automorphism groups of connected bipartite irreducible graphs

Let $G=(V,E)$ be a graph with the vertex-set $V$ and the edge-set $E$. Let $N(v)$ denote the set of neighbors of the vertex $v$ of $G.$ The graph $G$ is called $ irreducible $ whenever for every $v,w \in V$ if $v \neq w$, then $N(v)\neq N(w).$ In this paper, we present a method for finding automorphism groups of connected bipartite irreducible graphs. Then, by our method, we determine automorphism groups of some classes of connected bipartite irreducible graphs, including a class of graphs which are derived from Grassmann graphs. Let $a_0$ be a fixed positive integer. We show that if $G$ is a connected non-bipartite irreducible graph such that $c(v,w)=|N(v)\cap N(w)|=a_0$ when $v,w$ are adjacent, whereas $c(v,w) \neq a_0$, when $v,w$ are not adjacent, then $G$ is a $stable$ graph, that is, the automorphism group of the bipartite double cover of $G$ is isomorphic with the group $Aut(G) \times \mathbb{Z}_2$. Finally, we show that the Johnson graph $J(n,k)$ is a stable graph.

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arXiv2019

Johnson graphs are panconnected

For any given $n,m \in \mathbb{N}$ with $ m < n $, the Johnson graph $J(n,m)$ is defined as the graph whose vertex set is $V=\{v\mid v\subseteq [n]=\{1,...,n\}, |v|=m\}$, where two vertices $v$,$w$ are adjacent if and only if $|v\cap w|=m-1$. A graph $G$ of order $n > 2$ is panconnected if for every two vertices $u$ and $v$, there is a $u$-$v$ path of length $l$ for every integer $l$ with $d(u,v) \leq l \leq n-1$. In this paper, we prove that the Johnson graph $J(n,m)$ is a panconnected graph.

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arXiv2019

Cayley properties of the line graphs induced by consecutive layers of the hypercube

Let $n >3$ and $ 0< k < \frac{n}{2} $ be integers. In this paper, we investigate some algebraic properties of the line graph of the graph $ {Q_n}(k,k+1) $ where $ {Q_n}(k,k+1) $ is the subgraph of the hypercube $Q_n$ which is induced by the set of vertices of weights $k$ and $k+1$. In the first step, we determine the automorphism groups of these graphs for all values of $n,k$. In the second step, we study Cayley properties of the line graphs of these graphs. In particular, we show that if $k\geq 3$ and $ n \neq 2k+1$, then except for the cases $k=3, n=9$ and $k=3, n=33$, the line graph of the graph $ {Q_n}(k,k+1) $ is a vertex-transitive non-Cayley graph. Also, we show that the line graph of the graph $ {Q_n}(1,2) $ is a Cayley graph if and only if $ n$ is a power of a prime $p$. Moreover, we show that for \lq{}almost all\rq{} even values of $k$, the line graph of the graph $ {Q_{2k+1}}(k,k+1) $ is a vertex-transitive non-Cayley graph.

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arXiv2019

A note on the automorphism group of the Hamming graph

Let $Ω$ be a $m$-set, where $m>1$, is an integer. The Hamming graph $H(n,m)$, has $Ω^{n}$ as its vertex-set, with two vertices are adjacent if and only if they differ in exactly one coordinate. In this paper, we provide a proof on the automorphism group of the Hamming graph $H(n,m)$, by using elementary facts of group theory and graph theory.

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arXiv2018

The automorphism group of the bipartite Kneser graph

Let $n$ and $k$ be integers with $n>2k, k\geq1$. We denote by $H(n, k)$ the $bipartite\ Kneser\ graph$, that is, a graph with the family of $k$-subsets and ($n-k$)-subsets of $[n] = \{1, 2, ... , n\}$ as vertices, in which any two vertices are adjacent if and only if one of them is a subset of the other. In this paper, we determine the automorphism group of $H(n, k)$. We show that $Aut(H(n, k))\cong Sym([n]) \times \mathbb{Z}_2$ where $\mathbb{Z}_2$ is the cyclic group of order $2$. Then, as an application of the obtained result, we give a new proof for determining the automorphism group of the Kneser graph $K(n,k)$. In fact we show how to determine the automorphism group of the Kneser graph $K(n,k)$ given the automorphism group of the Johnson graph $J(n,k)$. Note that the known proofs for determining the automorphism groups of Johnson graph $J(n,k)$ and Kneser graph $ K(n,k)$ are independent from each other.

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arXiv2018

Some algebraic properties of bipartite Kneser graphs

Let $n$ and $k$ be integers with $n> k\geq1$ and $[n] = \{1, 2, ... , n\} $. The $bipartite \ Kneser \ graph$ $H(n, k)$ is the graph with the all $k$-element and all ($n-k$)-element subsets of $[n] $ as vertices, and there is an edge between any two vertices, when one is a subset of the other. In this paper, we show that $H(n, k)$ is an arc-transitive graph. Also, we show that $H(n,1)$ is a distance-transitive Cayley graph. Finally, we determine the automorphism group of the graph $H(n, 1)$ and show that $Aut(H(n, 1)) \cong Sym([n] )$ $\times \mathbb{Z}_2$, where $\mathbb{Z}_2$ is the cyclic group of order $2$. Moreover, we pose some open problems about the automorphism group of the bipartite Kneser graph $H(n, k)$.

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arXiv2017

A class of integral graphs constructed from the hypercube

In this paper, we determine the set of all distinct eigenvalues of the line graph which is induced by the first and second layers of the hypercube $ Q_n $, $n>3$. We show that this graph has precisely five distinct eigenvalues and all of its eigenvalues are integers

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