Related papers: Spectral properties of the $n$-Queens' Graphs

We show that all scaling quantum graphs are explicitly integrable, i.e. any one of their spectral eigenvalues $E_n$ is computable analytically, explicitly, and individually for any given $n$. This is surprising, since quantum graphs are…

This paper gives a tight upper bound on the spectral radius of the signless Laplacian of graphs of given order and clique number. More precisely, let G be a graph of order n, let A be its adjacency matrix, and let D be the diagonal matrix…

Let $G$ be a graph of order $n,$ and let $q_{1}(G) \geq ...\geq q_{n}(G) $ be the eigenvalues of the $Q$-matrix of $G$, also known as the signless Laplacian of $G.$ In this paper we give a necessary and sufficient condition for the equality…

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…

Let $q=p^e$, where $p$ is a prime and $e\geq 1$ is an integer. For $m\geq 1$, let $P$ and $L$ be two copies of the $(m+1)$-dimensional vector spaces over the finite field $\mathbb{F}_q$. Consider the bipartite graph $W_m(q)$ with partite…

We consider a problem proposed by Linial and Wilf to determine the structure of graphs that allows the maximum number of $q$-colorings among graphs with $n$ vertices and $m$ edges. Let $T_r(n)$ denote the Tur\'{a}n graph - the complete…

Given a graph $G$, we have the adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$. The $Q$-spectrum is the all eigenvalues of $Q$-matrix $Q(G)=A(G)+D(G)$. A class of graphs is determined by their generalized $Q$-spectrum (DGQS for…

The spectral radius $\rho(G)$ of a graph $G$ is the largest eigenvalue of its adjacency matrix $A(G)$. For a fixed integer $e\ge 1$, let $G^{min}_{n,n-e}$ be a graph with minimal spectral radius among all connected graphs on $n$ vertices…

We study the statistical properties of the scattering matrix associated with generic quantum graphs. The scattering matrix is the quantum analogue of the classical evolution operator on the graph. For the energy-averaged spectral form…

We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…

We introduce a two player game on an n x n chessboard where queens are placed by alternating turns on a chessboard square whose availability is determined by the number of queens already on the board which can attack that square modulo two.…

Recently the collection $\cal G$ of all signed graphs for which the adjacency matrix has all but at most two eigenvalues equal to $\pm 1$ has been determined. Here we investigate $\cal G$ for cospectral pairs, and for signed graphs…

The energy of a graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of $G$. Let $n$ be an even number and $\mathbb{U}_{n}$ be the set of all conjugated unicyclic graphs of order $n$ with maximum…

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 problem of multiway partitioning of an undirected graph is considered. A spectral method is used, where the k > 2 largest eigenvalues of the normalized adjacency matrix (equivalently, the k smallest eigenvalues of the normalized graph…

Using modular arithmetic of the ring $\mathbb{Z}_{n+1}$ we obtain a new short solution to the problem of existence of at least one solution to the $N$-Queens problem on an $N \times N$ chessboard. It was proved, that these solutions can be…

Let $G$ be an undirected graph on $n$ vertices and let $S(G)$ be the set of all $n \times n$ real symmetric matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of $G$. The inverse eigenvalue…

Graphs with few distinct eigenvalues have been investigated extensively. In this paper, we focus on another relevant topic: characterizing graphs with some eigenvalue of large multiplicity. Specifically, the normalized Laplacian matrix of a…

We propose a Nordhaus-Gaddum conjecture for $q(G)$, the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph $G$: for every graph $G$ excluding four exceptions, we conjecture that $q(G)+q(G^c)\le |G|+2$,…

Random planar graphs have been the subject of much recent work. Many basic properties of the standard uniform random planar graph P_{n}, by which we mean a graph chosen uniformly at random from the set of all planar graphs with vertex set…