Related papers: Eccentricity Spectra and Integral Eigenvalues of Z…
The distance matrix of a graph $G$ is the matrix containing the pairwise distances between vertices. The distance eigenvalues of $G$ are the eigenvalues of its distance matrix and they form the distance spectrum of $G$. We determine the…
The total reciprocal edge-eccentricity is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and…
Consider a semigraph $G=(V,\,E)$; in this paper, we study the eigenvalues of the Laplacian matrix of $G$. We show that the Laplacian of $G$ is positive semi-definite, and $G$ is connected if and only if $\lambda_2 >0.$ Along the similar…
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…
We consider unitary graphs attached to Z_d^n using an analogue of the Euclidean distance. These graphs are shown to be integral when n is odd or the dimension d is even.
Let $G$ be a finite, connected graph. The eccentricity of a vertex $v$ of $G$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is the arithmetic mean of the eccentricities of the vertices of $G$. We…
This article deals with the spectra of Laplacians of weighted graphs. In this context, two objects are of fundamental importance for the dynamics of complex networks: the second eigenvalue of such a spectrum (called algebraic connectivity)…
In this article, we relate the spectrum of the discrete magnetic Laplacian (DML) on a finite simple graph with two structural properties of the graph: the existence of a perfect matching and the existence of a Hamiltonian cycle of the…
A graph is non-trivial if it contains at least one nonloop edge. The essential connectivity of $G$, denoted by $\kappa'(G)$, is the minimum number of vertices of $G$ whose removal produces a disconnected graph with at least two components…
All graphs considered are simple and undirected. The Inverse Eigenvalue Problem of a Graph $G$ (IEP-G) aims to find all possible spectra for matrices whose $(i,j)-$entry, for $i\neq j$, is nonzero precisely when $i$ is adjacent to $j$. A…
The relation between the Wiener index $W(G)$ and the eccentricity $\varepsilon(G)$ of a graph $G$ is studied. Lower and upper bounds on $W(G)$ in terms of $\varepsilon(G)$ are proved and extremal graphs characterized. A Nordhaus-Gaddum type…
We studied how eccentricity affects the gravitational wave (GW) spectrum from supermassive black hole (SMBH) binaries. We developed a fast and accurate semi-analytic method for computing the GW spectra, the distribution for the spectral…
Spectral hypergraph theory has recently attracted considerable interest as it provides a natural framework for modeling higher-order relationships beyond classical graphs. In this setting, eigenvalues of adjacency, Laplacian, and…
The $A_{\alpha}$-matrix of a graph $G$ is the convex linear combination of the adjacency matrix $A(G)$ and the diagonal matrix of vertex degrees $D(G)$, i.e., $A_{\alpha}(G) = \alpha D(G) + (1 - \alpha)A(G)$, where $0\leq\alpha \leq1$. The…
Let $G$ be a digraph with adjacency matrix $A(G)$. Let $D(G)$ be the diagonal matrix with outdegrees of vertices of $G$. Nikiforov \cite{Niki} proposed to study the convex combinations of the adjacency matrix and diagonal matrix of the…
We study numerically the spectrum and eigenstate properties of the Google matrix of various examples of directed networks such as vocabulary networks of dictionaries and university World Wide Web networks. The spectra have gapless structure…
Given a finite simple graph $G$ on $m$ vertices, the zeon combinatorial Laplacian $\Lambda$ of $G$ is an $m\times m$ graph having entries in the complex zeon algebra $\mathbb{C}\mathfrak{Z}$. It is shown here that if the graph has a unique…
Inspired by a very recent work of A. {\DH}uri\'c, S. Jev{\dj}eni\'c and N. Stopar, we introduce a new definition of zero-divisor graphs attached to rings, that includes all of the classical definitions already known in the literature. We…
In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known…
We develop a basic theory for divisible design graphs with possible selfloops (LDDG's), and describe two infinite families of such graphs, some members of which are also classical examples of divisible design graphs without loops (DDG's).…