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Two graphs $G$ and $H$ are \emph{cospectral} if the adjacency matrices share the same spectrum. Constructing cospectral non-isomorphic graphs has been studied extensively for many years and various constructions are known in the literature,…
We use the line digraph construction to associate an orthogonal matrix with each graph. From this orthogonal matrix, we derive two further matrices. The spectrum of each of these three matrices is considered as a graph invariant. For the…
The cochromatic number $\zeta(G)$ of a graph $G$ is the smallest number of colors in a vertex-coloring of $G$ such that every color class forms an independent set or a clique. In three papers written around 1990, Erd\H{o}s, Gimbel and…
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…
Properties of graphs that can be characterized by the spectrum of the adjacency matrix of the graph have been studied systematically recently. Motivated by the complexity of these properties, we show that there are such properties for which…
A decomposition of a non-empty simple graph $G$ is a pair $[G,P]$, such that $P$ is a set of non-empty induced subgraphs of $G$, and every edge of $G$ belongs to exactly one subgraph in $P$. The chromatic index $\chi'([G,P])$ of a…
Given an edge-coloring of a graph $G$, we associate to every vertex $v$ of $G$ the set of colors appearing on the edges incident with $v$. The palette index of $G$ is defined as the minimum number of such distinct sets, taken over all…
This paper establishes an upper bound on the number of generalized cospectral mates of simple graphs, where the generalized spectrum consists of the spectrum of a graph and its complement. Moving beyond the classical problem of identifying…
Characterizing graphs by their spectra is an important topic in spectral graph theory, which has attracted a lot of attention of researchers in recent years. It is generally very hard and challenging to show a given graph to be determined…
Two graphs having the same spectrum are said to be cospectral. A pair of singularly cospectral graphs is formed by two graphs such that the absolute values of their nonzero eigenvalues coincide. Clearly, a pair of cospectral graphs is also…
We investigate the notion of quantum chromatic number of a graph, which is the minimal number of colours necessary in a protocol in which two separated provers can convince an interrogator with certainty that they have a colouring of the…
The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that there exists a $k$-vertex coloring of $G$ in which any two vertices receiving color $i$ are at distance at least $i+1$. It is proved that in…
A mixed graph has a set of vertices, a set of undirected egdes, and a set of directed arcs. A proper coloring of a mixed graph $G$ is a function $c$ that assigns to each vertex in $G$ a positive integer such that, for each edge $uv$ in $G$,…
A proper coloring of a graph $G$ is said to be a strong odd coloring of $G$, if for every vertex $v$ and every color $c$, either $c$ appears on an odd number of vertices in the neighborhood of $v$ or $c$ is absent in the neighborhood of…
The orbital bivariate chromatic polynomial, introduced in this article, counts the number of ways to color the vertices of a graph with $\lambda$ colors such that adjacent vertices either receive distinct colors from a set of $\lambda$…
A graph $H$ is said to be common if the number of monochromatic labelled copies of $H$ in a red/blue edge colouring of a large complete graph is asymptotically minimized by a random colouring with an equal proportion of each colour. We…
A graph $G$ is called chromatic-choosable if $\chi(G)=ch(G)$. A natural problem is to determine the minimum number of vertices in a $k$-chromatic non-$k$-choosable graph. It was conjectured by Ohba, and proved by Noel, Reed and Wu that…
The packing chromatic number $\chi_{\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $\Pi_1,\ldots,\Pi_k$, where $\Pi_i$, $i\in [k]$, is an $i$-packing. The following…
Several researchers have recently explored various graph parameters that can or cannot be characterized by the spectrum of a matrix associated with a graph. In this paper we show that several NP-hard zero forcing numbers are not…
The orthogonal rank of a graph $G=(V,E)$ is the smallest dimension $\xi$ such that there exist non-zero column vectors $x_v\in\mathbb{C}^\xi$ for $v\in V$ satisfying the orthogonality condition $x_v^\dagger x_w=0$ for all $vw\in E$. We…