Related papers: Equitable 2-partitions of the Hamming graphs with …

Graph operations or products play an important role in complex networks. In this paper, we study the properties of $q$-subdivision graphs, which have been applied to model complex networks. For a simple connected graph $G$, its…

Suppose we are given two graphs on $n$ vertices. We define an observable in the Hilbert space $\Co[(S_n \wr S_2)^m]$ which returns the answer ``yes'' with certainty if the graphs are isomorphic and ``no'' with probability at least…

In this note, we consider connected graphs with exactly two main eigenvalues. We will give several constructions for them, and as a consequence we show a family of those graphs with an unbounded number of distinct valencies.

We compute the dimension $d_{n,r}(q) = \dim(\IR_q^r)$ of the defining module $\IR_q^r$ for the $q$-partition algebra. This module comes from $r$-iterations of Harish-Chandra restriction and induction on $\GL_n(\FF_q)$. This dimension is a…

Let $n\geq 2$ be an integer, and let $i\in\{0,...,n-1\}$. An $i$-th dimension edge in the $n$-dimensional hypercube $Q_n$ is an edge ${v_1}{v_2}$ such that $v_1,v_2$ differ just at their $i$-th entries. The parity of an $i$-th dimension…

Identifying codes in graphs have been widely studied since their introduction by Karpovsky, Chakrabarty and Levitin in 1998. In particular, there are a lot of results regarding the binary hypercubes, that is, the Hamming graphs $K_2^n$. In…

A partition of degree $n$ is a decomposition $n=i_1+i_2+\dots+i_q$, where ${i_1,i_2,\dots,i_q}$ are positive integers called the parts of the partition. Let $\lambda>0$ be an integer. The partition is said to be a $\lambda$--partition if…

N.Wallach has considered an element of the group algebra of the symmetric group S_n which is the sum of an n-cycle, an (n-1)-cycle,...,a 2-cycle and the identity. He showed that multiplication by this element has eigenvalues…

For each positive integer $n$, the Fibonacci-sum graph $G_n$ on vertices $1,2,\ldots,n$ is defined by two vertices forming an edge if and only if they sum to a Fibonacci number. It is known that each $G_n$ is bipartite, and all Hamiltonian…

A graph $G$ is $q$-Ramsey for another graph $H$ if in any $q$-edge-colouring of $G$ there is a monochromatic copy of $H$, and the classic Ramsey problem asks for the minimum number of vertices in such a graph. This was broadened in the…

We give lower and upper bounds on the burning number of Hamming graphs, Johnson graphs, and halved cube graphs. For the lower bounds, we use the fact that $1$-skeletons of the eigenpolytopes of these graphs are isomorphic to the original…

A number of recent papers have considered signed graph Laplacians, a generalization of the classical graph Laplacian, where the edge weights are allowed to take either sign. In the classical case, where the edge weights are all positive,…

This article focuses on finding the eigenvalues of the Laplacian matrix of the comaximal graph $\Gamma(\mathbb Z_n)$ of the ring $\mathbb Z_n$ for $n> 2$. We determine the eigenvalues of $\Gamma(\mathbb Z_n)$ for various $n$ and also…

We consider a 2-homogeneous bipartite distance-regular graph $\Gamma$ with diameter $D \geq 3$. We assume that $\Gamma$ is not a hypercube nor a cycle. We fix a $Q$-polynomial ordering of the primitive idempotents of $\Gamma$. This…

We study perfect $2$-coloring of the Johnson graphs $J(n,3)$ associated with the third largest eigenvalue and symmetric quotient matrix, which exists only when $n \in \{6, 10\}$. We survey the known constructions in the case $n=6$, give a…

Let $\Omega$ be a $m$-set, where $m>1$, is an integer. The Hamming graph $H(n,m)$, has $\Omega ^{n}$ 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…

In this paper, we present a method to obtain regular (or equitable) partitions of Cayley (di)graphs (that is, graphs, digraphs, or mixed graphs) of permutation groups on $n$ letters. We prove that every partition of the number $n$ gives…

The $q$-analogue of the binomial coefficient, known as a $q$-binomial coefficient, is typically denoted $\left[{n \atop k}\right]_q$. These polynomials are important combinatorial objects, often appearing in generating functions related to…

Let $q$ be a positve integer, and $G$ be a $q$-partite simple graph on $qn$ vertices, with $n$ vertices in each vertex class. Let $\delta={k_q \over k_q+1}$, where $k_q=q+O(\log{q})$. If each vertex of $G$ is adjacent to at least $\delta n$…

The commutation relations between the generalized Pauli operators of N-qudits (i. e., N p-level quantum systems), and the structure of their maximal sets of commuting bases, follow a nice graph theoretical/geometrical pattern. One may…