Related papers: Graphs with three distinct distance eigenvalues

Let $\eta_{1}\ge \eta_{2}\ge\cdots\ge \eta_{n}$ be the eigenavalues of $\mathcal{ABS}$ matrix. In this paper, we characterize connected graphs with $\mathcal{ABS}$ eigenvalue $\eta_{n}>-1$. As a result, we determine all connected graphs…

Suppose $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d_G( v_i,v_j ) $ be the least distance between $v_i$ and $v_j$ in $G$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} ) _{n\times…

Suppose that $G$ is a connected simple graph with the vertex set $V( G ) = \{ v_1,v_2,\cdots ,v_n \} $. Let $d( v_i,v_j ) $ be the distance between $v_i$ and $v_j$. Then the distance matrix of $G$ is $D( G ) =( d_{ij} )_{n\times n}$, where…

In this paper we study when the $q$-distance matrix of a distance-regular graph has few distinct eigenvalues. We mainly concentrate on diameter 3.

We introduce a new model of indeterminacy in graphs: instead of specifying all the edges of the graph, the input contains all triples of vertices that form a connected subgraph. In general, different (labelled) graphs may have the same set…

For a graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A\in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ be the minimum number…

For a graph $G$, let $L(G)$ and $Q(G)$ be the Laplacian and signless Laplacian matrices of $G$, respectively, and $\tau(G)$ be the number of spanning trees of $G$. We prove that if $G$ has an odd number of vertices and $\tau(G)$ is not…

Let $D(G)$ denote the distance matrix of a connected graph $G$ with $n$ vertices. The distance spectral gap of a graph $G$ is defined as $\delta_{D^G} = \rho_1 - \rho_2$, where $\rho_1$ and $\rho_2$ represent the largest and second largest…

We study regular graphs whose distance-$2$ graph or distance-$1$-or-$2$ graph is strongly regular. We provide a characterization of such graphs $\Gamma$ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the…

The \textit{eccentricity matrix} $\mathcal{E}(G)$ of a connected graph $G$ is obtained from the distance matrix of $G$ by keeping the largest non-zero entries in each row and each column, and leaving zeros in the remaining ones. The…

We analyze graphs attaining the extreme values of various spectral indices in the class of all simple connected graphs, as well as in the class of graphs which are not complete multipartite graphs. We also present results on density of…

The distance energy of a simple connected graph $G$ is defined as the sum of absolute values of its distance eigenvalues. In this paper, we mainly give a positive answer to a conjecture of distance energy of clique trees proposed by Lin,…

In this paper, we characterize all graphs with eigenvectors of the signless Laplacian and adjacency matrices with components equal to $\{- 1, 0, 1\}.$ We extend the graph parameter max $k$-cut to square matrices and prove a general sharp…

The distance energy of a graph $G$ is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of $G$. There was a vast research for the pairs and families of non-cospectral…

Let $k\ge 2$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $\delta$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with…

In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of…

We establish new bounds on the minimum number of distinct eigenvalues among real symmetric matrices with nonzero off-diagonal pattern described by the edges of a graph and apply these to determine the minimum number of distinct eigenvalues…

A cograph is a simple graph which contains no path on 4 vertices as an induced subgraph. We consider the eigenvalues of adjacency matrices of cographs and prove that a graph $G$ is a cograph if and only if no induced subgraph of $G$ has an…

We provide a finite equational presentation of graphs of treewidth at most three, solving an instanceof an open problem by Courcelle and Engelfriet. We use a syntax generalising series-parallel expressions, denoting graphs with a small…

There are several interrelated notions of discrete curvature on graphs. Many approaches utilize the optimal transportation metric on its probability simplex or the distance matrix of the graph. In this survey article, we compute formulas…