Related papers: The graphs with exactly two distance eigenvalues d…

The maximum independent sets in the Doob graphs D(m,n) are analogs of the distance-2 MDS codes in Hamming graphs and of the latin hypercubes. We prove the characterization of these sets stating that every such set is semilinear or…

We introduce the concept of distance ideals of graphs, which can be regarded as a generalization of the Smith normal form and the spectra of the distance matrix of a graph. We obtain a classification of the graphs with at most one trivial…

Sharp bounds on the least eigenvalue of an arbitrary graph are presented. Necessary and sufficient (just sufficient) conditions for the lower (upper) bound to be attained are deduced using edge clique partitions. As an application, we prove…

Given a finite group $G$, the character graph, denoted by $\Delta(G)$, for its irreducible character degrees is a graph with vertex set $\rho(G)$ which is the set of prime numbers that divide the irreducible character degrees of $G$, and…

For non-negative integers~$k$, we consider graphs in which every vertex has exactly $k$ vertices at distance~$2$, i.e., graphs whose distance-$2$ graphs are $k$-regular. We call such graphs $k$-metamour-regular motivated by the terminology…

We study the metric dimension (strong and weak) of infinite graphs. In particular, our main interest is characterizing infinite graphs with finite dimension. Our main results: (1) graphs with more than one end have infinite strong…

For $0\le \alpha\le 1$, Nikiforov proposed to study the spectral properties of the family of matrices $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$ of a graph $G$, where $D(G)$ is the degree diagonal matrix and $A(G)$ is the adjacency matrix.…

The sum of the absolute values of the eigenvalues of a graph is called the energy of the graph. We study the problem of finding graphs with extremal energy within specified classes of graphs. We develop tools for treating such problems and…

Let $G$ be a simple graph and $A(G)$ be the adjacency matrix of $G$. The matrix $S(G) = J -I -2A(G)$ is called the Seidel matrix of $G$, where $I$ is an identity matrix and $J$ is a square matrix all of whose entries are equal to 1.…

Bollob\'as and Nikiforov [J. Combin. Theory, Ser. B. 97 (2007) 859--865] conjectured the following. If $G$ is a $K_{r+1}$-free graph on at least $r+1$ vertices and $m$ edges, then $\lambda^2_1(G)+\lambda^2_2(G)\leq \frac{r-1}{r}\cdot2m$,…

Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be…

In this paper, connections between independent sets and the variety of mutual-visibility sets are studied. It is proved that every outer mutual-visibility set of a graph is independent if and only if the graph is distance edge-critical.…

Eigenvalues of a graph are of high interest in graph analytics for Big Data due to their relevance to many important properties of the graph including network resilience, community detection and the speed of viral propagation. Accurate…

Graph is an abstract representation commonly used to model networked systems and structure. In problems across various fields, including computer vision and pattern recognition, and neuroscience, graphs are often brought into comparison (a…

Distance-hereditary graphs form an important class of graphs, from the theoretical point of view, due to the fact that they are the totally decomposable graphs for the split-decomposition. The previous best enumerative result for these…

Let $n$ be any positive integer, the friendship graph $F_n$ consist of $n$ edge-disjoint triangles that all of them meeting in one vertex. A graph $G$ is called cospectral with a graph $H$ if their adjacency matrices have the same…

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…

Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup…

It is proved that the median eigenvalues of every connected bipartite graph $G$ of maximum degree at most three belong to the interval $[-1,1]$ with a single exception of the Heawood graph, whose median eigenvalues are $\pm\sqrt{2}$.…

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…