The spanning $k$-hypertree and adjacency spectral radius in hypergraphs

THE SPANNING K -HYPERTREE AND ADJACENCY SPECTRAL RADIUS IN HYPERGRAPHS ∗

Threshold Graphs Allow Few Distinct Eigenvalues: A New Approach

For any graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A \in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ represent the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we p...

Extremal Graphs with Prescribed Parameters for the Spectral Radius of Weighted Adjacency Matrices with Property $P^{\ast}$

In the literature, several graph matrices based on vertex degrees have been introduced, and the study of their spectral properties has attracted considerable attention in recent years. Motivated by these developments, in this paper, we investigate the spectral radius of the weighted adjacency matrix $A_f(G)$, where the...

Product approximations for uniform spanning trees and determinantal processes

We prove that for every finite connected simple graph $G$ on $n$ vertices with minimum degree $k\geq 2$, there is a coupling of its uniform spanning tree and its random $1$-out subgraph under which the expected number of differing edges is $O(n\log (k)/\sqrt{k})$. More generally, we establish analogous couplings betwee...

The Spectral Theory of Threshold Graphs: A Survey

Threshold graphs-the $\{P_4, C_4, 2K_2\}$-free graphs, equivalently those built from a single vertex by repeatedly adding isolated or dominating vertices-are among the most thoroughly studied graph classes in spectral graph theory. This survey collects results on the spectrum of threshold graphs across the full range o...

Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph

Let $S_k(G)$ denote the sum of the $k$ largest eigenvalues of a graph $G$. Motivated by the classical Hoffman program for the spectral radius of a graph, we investigate an additive Hoffman-type problem for $S_k(G)$. For each fixed $k\geq 2$ and sufficiently large order $n$, we characterize all connected graphs satisfyi...