Related papers: Doubling constants and spectral theory on graphs

Given a connected graph $G$, the metric (resp. edge metric) dimension of $G$ is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of $G$ by means of distance…

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph…

In this paper, a new measurement to compare two large-scale graphs based on the theory of quantum probability is proposed. An explicit form for the spectral distribution of the corresponding adjacency matrix of a graph is established. Our…

Let $r\geq 3$ be an integer and $G$ be a graph. Let $\delta(G), \Delta(G)$, $\alpha(G)$ and $\mu(G)$ denotes minimum degree, maximum degree, independence number and matching number of $G$, respectively. Recently, Caro, Davila and Pepper…

A dominating set in a graph $G$ is a subset of vertices $D$ such that every vertex in $V\setminus D$ is a neighbor of some vertex of $D$. The domination number of $G$ is the minimum size of a dominating set of $G$ and it is denoted by…

Let $G$ be a simple graph with $n$ vertices and $\pm 1$-weights on edges. Suppose that for every edge $e$ the sum of edges adjacent to $e$ (including $e$ itself) is positive. Then the sum of weights over edges of $G$ is at least…

The thickness $\theta(G)$ of a graph $G$ is the minimum number of planar spanning subgraphs into which the graph $G$ can be decomposed. As a topological invariant of a graph, it is a measurement of the closeness to planarity of a graph, and…

In this paper we give a new characterization of the dual polar graphs, extending the work of Brouwer and Wilbrink on regular near polygons. Also as a consequence of our characterization we confirm a conjecture of the authors on…

We study the average number $\mathcal{A}(G)$ of colors in the non-equivalent colorings of a graph $G$. We show some general properties of this graph invariant and determine its value for some classes of graphs. We then conjecture several…

In recent years, considerable advances have been made in the study of properties of metric spaces in terms of their doubling dimension. This line of research has not only enhanced our understanding of finite metrics, but has also resulted…

Let $d\geq 3$ be a fixed integer and $A$ be the adjacency matrix of a random $d$-regular directed or undirected graph on $n$ vertices. We show there exist constants $\mathfrak d>0$, \begin{align*} {\mathbb P}(\text{$A$ is singular in…

Over the past half century, the rigidity of graphs in $R^2$ has aroused a great deal of interest. Lov\'{a}sz and Yemini (1982) proved that every $6$-connected graph is rigid in $R^2$. Jackson and Jord\'{a}n (2005) provided a similar…

Let $\log^Cn\le d\le n/2$ for a sufficiently large constant $C>0$ and let $A_n$ denote the adjacency matrix of a uniform random $d$-regular directed graph on $n$ vertices. We prove that as $n$ tends to infinity, the empirical spectral…

A spectral characterization of the matching number (the size of a maximum matching) of a graph is given. More precisely, it is shown that the graphs G of order n whose matching number is k are precisely those graphs with the maximum skew…

For a given graph G and an associated class of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in G, the collection of all possible spectra for such matrices is considered. Building on the pioneering work…

For a graph $G$ with adjacency matrix $A(G)$ and degree diagonal matrix $D(G)$, the $A_{\alpha}$-matrix of $G$ is defined as \begin{equation*} A_{\alpha}(G) = \alpha D(G) + (1- \alpha) A(G), \text{ for any } \alpha \in [0,1].…

In this paper, we will give a structure theory for signed graphs with fixed smallest eigenvalue and investigate signed graphs with smallest eigenvalue greater than $-1-\sqrt{2}$. Given a real number $\lambda\leq -1$, we show that the…

A vertex coloring of a graph $G$ is called distinguishing if no non-identity automorphisms of $G$ can preserve it. The distinguishing number of $G$, denoted by $D(G)$, is the minimum number of colors required for such a coloring, and the…

A distinguishing coloring of a graph is a vertex coloring such that only the identity automorphism of the graph preserves the coloring. A 2-distinguishable graph is a graph which can be distinguished using 2 colors. The cost $\rho(G)$ of a…

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…