Related papers: The Perron non-backtracking eigenvalue after node …

Let F(G) be a fixed linear combination of the k extremal eigenvalues of a graph G and of its complement. The problem of finding max{F(G):v(G)=n} generalizes a number of problems raised previously in the literature. We show that the limit…

An eigenvalue of a graph $G$ is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero. It is well known that a graph $G$ has exactly two main eigenvalues if and only if there exists a unique pair of…

Networks are often studied using the eigenvalues of their adjacency matrix, a powerful mathematical tool with a wide range of applications. Since in real systems the exact graph structure is not known, researchers resort to random graphs to…

A version of ``preferential attachment'' random graphs, corresponding to linear ``weights'' with random ``edge additions,'' which generalizes some previously considered models, is studied. This graph model is embedded in a continuous-time…

Large-scale graph data in real-world applications is often not static but dynamic, i. e., new nodes and edges appear over time. Current graph convolution approaches are promising, especially, when all the graph's nodes and edges are…

The eigenvalues of the Laplacian matrix for a class of directed graphs with both positive and negative weights are studied. First, a class of directed signed graphs is investigated in which one pair of nodes (either connected or not) is…

A non-backtracking walk on a graph, $H$, is a directed path of directed edges of $H$ such that no edge is the inverse of its preceding edge. Non-backtracking walks of a given length can be counted using the non-backtracking adjacency…

In this paper, we initiate the study of the inverse eigenvalue problem for probe graphs. A probe graph is a graph whose vertices are partitioned into probe vertices and non-probe vertices such that the non-probe vertices form an independent…

We obtain a lower bound on each entry of the principal eigenvector of a non-regular connected graph.

Graph neural networks are experiencing a surge of popularity within the machine learning community due to their ability to adapt to non-Euclidean domains and instil inductive biases. Despite this, their stability, i.e., their robustness to…

Let $G$ be a graph, and let $\lambda(G)$ denote the smallest eigenvalue of $G$. First, we provide an upper bound for $\lambda(G)$ based on induced bipartite subgraphs of $G$. Consequently, we extract two other upper bounds, one relying on…

The parameter $q(G)$ of an $n$-vertex graph $G$ is the minimum number of distinct eigenvalues over the family of symmetric matrices described by $G$. We show that all $G$ with $e(\overline{G}) = |E(\overline{G})| \leq \lfloor n/2 \rfloor…

In prioritization schemes, based on pairwise comparisons, such as the Analytical Hierarchy Process, it is necessary to extract a cardinal ranking vector from a reciprocal matrix that is unlikely to be consistent. It is natural to choose…

The eccentricity matrix $\epsilon(G)$, of a connected graph $G$ is obtained by retaining the maximum distance from each row and column of the distance matrix of $G$ and the other entries are assigned with 0. In this paper, we discuss the…

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…

Previously, the graph permanent was introduced as a single-valued invariant for graphs $G$ with $|E(G)| = k(|V(G)|-1)$ for some $k \in \mathbb{Z}_{>0}$. Herein, we construct the extended graph permanent, an infinite sequence for all graphs.…

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 $G$ 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…

In this paper, we investigate the eigenvalues of the Laplacian matrix of the "graph of graphs", in which cubic graphs of order n are joined together using Whitehead moves. Our work follows recent results from arXiv:2303.13923 , which…

For a graph $G$, the mean subtree order of $G$ is the average order of a subtree of $G$. In this note, we provide counterexamples to a recent conjecture of Chin, Gordon, MacPhee, and Vincent, that for every connected graph $G$ and every…