Related papers: On co-edge-regular graphs with 4 distinct eigenval…

We construct a family of maximal linklessly embeddable graphs on $n$ vertices and $3n-5$ edges for all $n\ge 10$, and another family on $n$ vertices and $m< \frac{25n}{12}-\frac{1}{4}$ edges for all $n\ge 13$. The latter significantly…

We prove two conjectures in spectral extremal graph theory involving the linear combinations of graph eigenvalues. Let $\lambda_1(G)$ be the largest eigenvalue of the adjacency matrix of a graph $G$, and $\bar{G}$ be the complement of $G$.…

In this paper we consider $r$-regular graphs $G$ that admit the vertex set partition such that one of the induced subgraphs is the join of an $s$-vertex clique and a $t$-vertex co-clique and represents a star complement for an eigenvalue…

Let $\mathcal{C}$ be a class of graphs closed under taking induced subgraphs. We say that $\mathcal{C}$ has the {\em clique-stable set separation property} if there exists $c \in \mathbb{N}$ such that for every graph $G \in \mathcal{C}$…

In his survey "Beyond graph energy: Norms of graphs and matrices" (2016), Nikiforov proposed two problems concerning characterizing the graphs that attain equality in a lower bound and in a upper bound for the energy of a graph,…

Motivated by discrete Laplacian differential operators with various accuracy orders in numerical analysis, we introduce new matrices attached to a simple graph that can be considered graph Laplacians with higher accuracy. In particular, we…

Jiang, Tidor, Yao, Zhang, and Zhao recently showed that connected bounded degree graphs have sublinear second eigenvalue multiplicity (always referring to the adjacency matrix). This result was a key step in the solution to the problem of…

Among the seven known (non-degenerate) triangle-free strongly regular graphs, we prove that the Clebsch graph describes a matrix with exactly two distinct eigenvalues while five of the graphs do not. In showing that the minimum number of…

A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper, $4$-valent one-regular graphs of order $5p^2$, where $p$ is a prime, are classified

We prove that after an arbitrarily small adjustment of edge lengths, the spectrum of a compact quantum graph with $\delta$-type vertex conditions can be simple. We also show that the eigenfunctions, with the exception of those living…

Let $\mathcal{G}$ be the set of simple graphs (or multigraphs) $G$ such that for each $G \in \mathcal{G}$ there exists at least two non-empty disjoint proper subsets $V_{1},V_{2}\subseteq V(G)$ satisfying $V(G)\setminus(V_{1} \cup…

The smallest eigenvalue of a graph is the smallest eigenvalue of its adjacency matrix. We show that the family of graphs with smallest eigenvalue at least $-\lambda$ can be defined by a finite set of forbidden induced subgraphs if and only…

Let $G$ be a regular graph with $m$ edges, and let $\mu_1, \mu_2$ denote the two largest eigenvalues of $A_G$, the adjacency matrix of $G$. We show that, if $G$ is not complete, then $$\mu_1^2 + \mu_2^2 \leq \frac{2(\omega - 1)}{\omega} m$$…

Answering a question of Jiang and Polyanskii as well as Jiang, Tidor, Yao, Zhang, and Zhao, we show the existence of infinitely many angles $\theta$ for which the maximum number of lines in $\mathbb R^n$ meeting at the origin with pairwise…

Recently Friedman proved Alon's conjecture for many families of d-regular graphs, namely that given any epsilon > 0 `most' graphs have their largest non-trivial eigenvalue at most 2 sqrt{d-1}+epsilon in absolute value; if the absolute value…

Liu, Hong, Gu, and Lai proved if the second largest eigenvalue of the adjacency matrix of graph $G$ with minimum degree $\delta \ge 2m+2 \ge 4$ satisfies $\lambda_2(G) < \delta - \frac{2m+1}{\delta+1}$, then $G$ contains at least $m+1$…

A set of graphs are called cospectral if their adjacency matrices have the same characteristic polynomial. In this paper we introduce a simple method for constructing infinite families of cospectral regular graphs. The construction is valid…

Let $G$ be a graph with adjacency matrix $A(G)$. We conjecture that \[2n^+(G) \le n^-(G)(n^-(G) + 1),\] where $n^+(G)$ and $n^-(G)$ denote the number of positive and negative eigenvalues of $A(G)$, respectively. This conjecture generalizes…

In this note, we present a structural description of certain connected cographs having $k \geq 2$ main signless Laplacian eigenvalues. This result allows us to characterize the cographs which are quasi-threshold graphs with two main…

For a connected graph $G$, we denote by $L(G)$, $m_{G}(\lambda)$, $c(G)$ and $p(G)$ the line graph of $G$, the eigenvalue multiplicity of $\lambda$ in $G$, the cyclomatic number and the number of pendant vertices in $G$, respectively. In…