Related papers: Geometric and spectral analysis on weighted digrap…

We consider a non self-adjoint Laplacian on a directed graph with non symmetric weights on edges. We give a criterion for the m-accretiveness and the m-sectoriality of this Laplacian. Our results are based on a comparison of this operator…

Associative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities:…

We study the interplay between spectrum, geometry and boundary conditions for two distinguished self-adjoint realisations of the Laplacian on infinite metric graphs, the so-called riedrichs and Neumann extensions. We introduce a new…

The monography examines the problem of constructing a group of automorphisms of a graph. A graph automorphism is a mapping of a set of vertices onto itself that preserves adjacency. The set of such automorphisms forms a vertex group of a…

Let $k$ 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 edge…

In the spirit of Arthur's trace formula, we establish a general trace formula for symmetric spaces associated with the variety of involutions of a finite $D$-module where $D$ is a division algebra central over a number field $F$. Such a…

Given a digraph $D=(V,A)$ on $n$ vertices and a vertex $v\in V$, the cycle-degree of $v$ is the minimum size of a set $S \subseteq V(D) \setminus \{v\}$ intersecting every directed cycle of $D$ containing $v$. From this definition of…

Let $G$ be a simple graph with adjacency matrix $A(G)$, signless Laplacian matrix $Q(G)$, degree diagonal matrix $D(G)$ and let $l(G)$ be the line graph of $G$. In 2017, Nikiforov defined the $A_\alpha$-matrix of $G$, $A_\alpha(G)$, as a…

The $\alpha$-Hermitian adjacency matrix $H_\alpha$ of a mixed graph $X$ has been recently introduced. It is a generalization of the adjacency matrix of unoriented graphs. In this paper, we consider a special case of the complex number…

We study the properties of several proximity measures for the vertices of weighted multigraphs and multidigraphs. Unlike the classical distance for the vertices of connected graphs, these proximity measures are applicable to weighted…

We study directed, weighted graphs $G=(V,E)$ and consider the (not necessarily symmetric) averaging operator $$ (\mathcal{L}u)(i) = -\sum_{j \sim_{} i}{p_{ij} (u(j) - u(i))},$$ where $p_{ij}$ are normalized edge weights. Given a vertex $i…

Random geometric graphs are a popular choice for a latent points generative model for networks. Their definition is based on a sample of $n$ points $X_1,X_2,\cdots,X_n$ on the Euclidean sphere~$\mathbb{S}^{d-1}$ which represents the latent…

Ahanjideh, Akbari, Fakharan and Trevisan proposed a conjecture in [Linear Algebra Appl. 632 (2022) 1--14] on the distribution of the Laplacian eigenvalues of graphs: for any connected graph of order $n$ with diameter $d\ge 2$ that is not a…

In this work, we analyze the Dirichlet Laplacian $-\Delta_{\Omega}^D$ in an unbounded waveguide $\Omega \subset \mathbb R^3$, where the cross-section is translated in a constant direction and rotated along a spatial line. We focus on the…

Let $\Omega \subset \mathbb R^3$ be a waveguide which is obtained by translating a cross-section in a constant direction along an unbounded spatial curve. Consider $-\Delta_{\Omega}^D$ the Dirichlet Laplacian operator in $\Omega$. Under the…

The \emph{simplicial rook graph} SR(d,n) is the graph whose vertices are the lattice points in the $n$th dilate of the standard simplex in $\mathbb{R}^d$, with two vertices adjacent if they differ in exactly two coordinates. We prove that…

In this paper we show that the $d$-dimensional algebraic connectivity of an arbitrary graph $G$ is bounded above by its $1$-dimensional algebraic connectivity, i.e., $a_d(G) \leq a_1(G)$, where $a_1(G)$ corresponds the well-studied second…

It is reported that dynamical systems over digraphs have superior performance in terms of system damping and tolerance to time delays if the underlying graph Laplacian has a purely real spectrum. This paper investigates the topological…

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…

The aim of this article is to give a simple geometric condition that guarantees the existence of spectral gaps of the discrete Laplacian on periodic graphs. For proving this, we analyse the discrete magnetic Laplacian (DML) on the finite…