Related papers: Two characterizations of the grid graphs
Let $G$ be a connected graph on $n$ vertices, and let $D(G)$ be the distance matrix of $G$. Let $\partial_1(G)\ge\partial_2(G)\ge\cdots\ge\partial_n(G)$ denote the eigenvalues of $D(G)$. In this paper, we characterize all connected 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…
We study oriented graphs whose Hermitian adjacency matrices of the second kind have few eigenvalues. We give a complete characterization of the oriented graphs with two distinct eigenvalues, showing that there are only four such graphs. We…
Combinatorial characterisations are obtained of symmetric and anti-symmetric infinitesimal rigidity for two-dimensional frameworks with reflectional symmetry in the case of norms where the unit ball is a quadrilateral and where the…
Among all simple nonbipartite 2-connected graphs and among all nonbipartite $\theta$-graphs, the minimum least $Q$-eigenvalues are completely determined, respectively.
We study covering graphs of the Paley graph associated to a finite field of characteristic p in the case where the covering transformation group is cyclic of prime order distinct from p. When the field has q = p elements, we show that the…
This survey paper contains a tutorial introduction to distance-regular graphs, with an emphasis on the subconstituent algebra and the $Q$-polynomial property.
The aim of this chapter is to provide an adequate graph theoretic framework for the description of periodic bifurcations which have recently been discovered in descendant trees of finite p-groups. The graph theoretic concepts of rooted…
An edge-colouring of a graph is distinguishing, if the only automorphism which preserves the colouring is the identity. It has been conjectured that all but finitely many connected, finite, regular graphs admit a distinguishing…
A graph is called a nut graph if zero is its eigenvalue of multiplicity one and its corresponding eigenvector has no zero entries. A graph is a bicirculant if it admits an automorphism with two equally sized vertex orbits. There are four…
We characterise the quartic (i.e. 4-regular) multigraphs with the property that every edge lies in a triangle. The main result is that such graphs are either squares of cycles, line multigraphs of cubic multigraphs, or are obtained from…
Graphs derived from groups are a widely studied class of graphs, motivated by their highly symmetric structure. In particular, G-graphs offer an easy and interesting alternative construction of semi-symmetric graphs. After recalling the…
For a graph $G$, let $\mathcal{S}(G)$ be the set consisting of Hermitian matrices whose graph is $G$. Denoted by $m_B(G,\lambda)$ the multiplicity of an eigenvalue $\lambda$ of $B(G)\in \mathcal{S}(G)$, we show that $m_B(G,\lambda)\le…
We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes…
We give a sharp bound for the automorphism group of a cubic simple graph with a given number of vertices. For each number of vertices we give an explicit graph attaining the bound, and prove its uniqueness in special cases.
Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to k-uniform hypergraphs, but only for…
We graph-theoretically characterize the class of graphs $G$ such that $I(G)^2$ are Buchsbaum.
In this paper we develop three characterizations for isomorphism of graphs. The first characterization is obtained by associating certain bitableaux with the graphs. We order these bitableaux by suitably defined lexicographic order and…
If $G$ is a graph, its Laplacian is the difference between diagonal matrix of its vertex degrees and its adjacency matrix. A one-edge connection of two graphs $G_{1}$ and $G_{2}$ is a graph $G=G_{1}\odot G_{2}$ with $V(G)=V(G_{1})\cup…
This paper deals with three graph characteristics related to graph covering named the (vertex, edge, and total, resp.) H-irregularity strength of a graph G admitting H-covering. Those are the minimum values of positive integer k such that G…