Related papers: New graph invariants based on $p$-Laplacian eigenv…

This note introduces a result on the location of eigenvalues, i.e., the spectrum, of the Laplacian for a family of undirected graphs with self-loops. We extend on the known results for the spectrum of undirected graphs without self-loops or…

In this paper, we study a nonlocal variational problem which consists of minimizing in $L^2$ the sum of a quadratic data fidelity and a regularization term corresponding to the $L^p$-norm of the nonlocal gradient. In particular, we study…

In 2017, Nikiforov introduced the concept of the $A_{\alpha}$-matrix, as a linear convex combination of the adjacency matrix and the degree diagonal matrix of a graph. This matrix has attracted increasing attention in recent years, as it…

We introduce and study Laplacians on a finite metric graph endowed with generalized densities, that is, measures of finite mass. One important motivation is that this setting provides a common framework for several interesting classes of…

For $k\ge 1$, the $k$-independence number $\alpha_k$ of a graph is the maximum number of vertices that are mutually at distance greater than $k$. The well-known inertia and ratio bounds for the (1-)independence number $\alpha(=\alpha_1)$ of…

Consider any random graph model where potential edges appear independently, with possibly different probabilities, and assume that the minimum expected degree is omega(ln n). We prove that the adjacency matrix and the Laplacian of that…

We show how the spectrum of a graph Laplacian changes with respect to a certain type of rank-one perturbation. We apply our finding to give new short proofs of the spectral version of Kirchhoff's Matrix Tree Theorem and known derivations…

We develop a nonlinear spectral graph theory, in which the Laplace operator is replaced by the 1-Laplacian ?$\Delta_1$. The eigenvalue problem is to solve a nonlinear system involving a set valued function. In the study, we investigate the…

We define a new graph invariant called the scramble number. We show that the scramble number of a graph is a lower bound for the gonality and an upper bound for the treewidth. Unlike the treewidth, the scramble number is not minor monotone,…

The structural properties of graphs are usually characterized in terms of invariants, which are functions of graphs that do not depend on the labeling of the nodes. In this paper we study convex graph invariants, which are graph invariants…

We consider a non self-adjoint Laplacian on a directed graph with non symmetric edge weights. We analyse spectral properties of this Laplacian under a Kirchhoff assumption. Moreover we establish isoperimet-ric inequalities in terms of the…

We study a nonlinear, nonlocal eigenvalue problem driven by the fractional p-Laplacian with an indefinite, singular weight chosen in an optimal class. We prove the existence of an unbounded sequence of positive variational eigenvalues and…

Eigenvalue spectrum of the Laplacian on a metric graph with arbitrary but fixed vertex conditions is investigated in the limit as the lengths of all edges decrease to zero at the same rate. It is proved that there are exactly four possible…

We examine diagonal combs, a recently identified class of infinite metric graphs whose properties depend on one parameter. These graphs exhibit a fascinating regime where they possess infinite volume while maintaining purely discrete…

On a compact metric graph, we consider the spectrum of the Laplacian defined with a mix of standard and Dirichlet vertex conditions. A Cheeger-type lower bound on the gap $\lambda_2 - \lambda_1$ is established, with a constant that depends…

We study the structure of eigenfunctions of the Laplacian on quantum graphs, with a particular focus on Morse eigenfunctions via nodal and Neumann domains. Building on Courant-type arguments, we establish upper bounds for the number of…

We address the problem of computing the graph $p$-Laplacian eigenpairs for $p\in (2,\infty)$. We propose a reformulation of the graph $p$-Laplacian eigenvalue problem in terms of a constrained weighted Laplacian eigenvalue problem and…

The von Neumann entropy of a graph is a spectral complexity measure that has recently found applications in complex networks analysis and pattern recognition. Two variants of the von Neumann entropy exist based on the graph Laplacian and…

We consider a magnetic Laplacian with periodic magnetic potentials on periodic discrete graphs. Its spectrum consists of a finite number of bands, where degenerate bands are eigenvalues of infinite multiplicity. We obtain a specific…

In this note we give a new upper bound for the Laplacian eigenvalues of an unweighted graph. Let $G$ be a simple graph on $n$ vertices. Let $d_{m}(G)$ and $\lambda_{m+1}(G)$ be the $m$-th smallest degree of $G$ and the $m+1$-th smallest…