Related papers: Some results on Hamming graphs and an extended Ham…

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…

Let $\Gamma=(V,E)$ be a graph. The square graph $\Gamma^2$ of the graph $\Gamma$ is the graph with the vertex set $V(\Gamma^2)=V$ in which two vertices are adjacent if and only if their distance in $\Gamma$ is at most two. The square graph…

The Gilbert graph $\text{Gilbert}(q,n,d)$, which arises naturally in graph theory and coding theory, is the regular graph on $\mathbb{F}_q^n$ in which two vertices are adjacent if their Hamming distance is less than $d$, and it is…

A Hamilton decomposition of a graph is a partitioning of its edge set into disjoint spanning cycles. The existence of such decompositions is known for all hypercubes of even dimension $2n$. We give a decomposition for the case $n = 2^a3^b$…

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…

We prove that any one-ended, locally finite Cayley graph with non-torsion generators admits a decomposition into edge-disjoint Hamiltonian (i.e. spanning) double-rays. In particular, the $n$-dimensional grid $\mathbb{Z}^n$ admits a…

In \cite{Chan95}, the authors classified the 2-extendable abelian Cayley graphs and posed the problem of characterizing all 2-extendable Cayley graphs. We first show that a connected bipartite Cayley (vertex-transitive) graph is…

The spined cube $SQ_n$ is a variant of the hypercube $Q_n$, introduced by Zhou et al. in [Information Processing Letters 111 (2011) 561-567] as an interconnection network for parallel computing. A graph $\G$ is an $m$-Cayley graph if its…

The Cayley graphs of finite groups are known to provide several examples of families of expanders, and some of them are Ramanujan graphs. Babai studied isospectral non-isomorphic Cayley graphs of the dihedral groups. Lubotzky, Samuels and…

We define and investigate a new three-parameter family of graphs that further generalizes the Fibonacci and metallic cubes. Namely, the number of vertices in this family of graphs satisfies Horadam recurrence, a linear recurrence of second…

In connection with his solution of the Sensitivity Conjecture, Hao Huang (arXiv: 1907.00847, 2019) asked the following question: Given a graph $G$ with high symmetry, what can we say about the smallest maximum degree of induced subgraphs of…

Let $\Gamma$ be a Cayley graph, or a Cayley sum graph, or a twisted Cayley graph, or a twisted Cayley sum graph, or a vertex-transitive graph. Suppose $\Gamma$ is undirected and non-bipartite. Let $\mu$ (resp. $\mu_2$) denote the smallest…

For a graph $G$, the spectral radius $\lambda_{1}(G)$ of $G$ is the largest eigenvalue of its adjacency matrix. An odd wheel $W_{2k+1}$ with $k\geq2$ is a graph obtained from a cycle of order $2k$ by adding a new vertex connecting to all…

It was proved in [Y.-Q. Feng, C. H. Li and J.-X. Zhou, Symmetric cubic graphs with solvable automorphism groups, {\em European J. Combin.} {\bf 45} (2015), 1-11] that a cubic symmetric graph with a solvable automorphism group is either a…

Let $G$ be a finite, undirected $d$-regular graph and $A(G)$ its normalized adjacency matrix, with eigenvalues $1 = \lambda_1(A)\geq \dots \ge \lambda_n \ge -1$. It is a classical fact that $\lambda_n = -1$ if and only if $G$ is bipartite.…

We say that two graphs $H_1,H_2$ on the same vertex set are $G$-creating ($G$-different in other papers, this difference is explained in the introduction) if the union of the two graphs contains $G$ as a subgraph. Let $H(n,k)$ be the…

Motivated to find the answers to some of the questions that have occurred in recent papers dealing with Hamiltonian cycles (abbreviated HCs) in some special classes of grid graphs we started the investigation of spanning unions of cycles,…

We construct a family of countexamples to a conjecture of Galvin [5], which stated that for any $n$-vertex, $d$-regular graph $G$ and any graph $H$ (possibly with loops), \[\hom(G,H) \leq \max\left\lbrace\hom(K_{d,d}, H)^{\frac{n}{2d}},…

The simple graphs $G=(V,E)$ that satisfy $|E'|\leq 2|V'|-l$ for any subgraph (and for $l=1,2,3$) are the $(2,l)$-sparse graphs. Those that also satisfy $|E|=2|V|-l$ are the $(2,l)$-tight graphs. These can be characterised by their…

Let $H$ and $G$ be two finite graphs. Define $h_H(G)$ to be the number of homomorphisms from $H$ to $G$. The function $h_H(\cdot)$ extends in a natural way to a function from the set of symmetric matrices to $\mathbb{R}$ such that for…