Related papers: The limit points of the top and bottom eigenvalues…

We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also…

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…

Let $G$ be a connected graph with vertex set $V$. The distance, $d_G(u, v)$, between vertices $u$ and $v$ of $G$ is defined as the length of a shortest path between $u$ and $v$ in $G$. The distance matrix of $G$ is the matrix $\mathbf{D}(G)…

For a graph $G$, let $f_2(G)$ denote the largest number of vertices in a $2$-regular subgraph of $G$. We determine the minimum of $f_2(G)$ over $3$-regular $n$-vertex simple graphs $G$. To do this, we prove that every $3$-regular multigraph…

In this paper we consider the degree/diameter problem, namely, given natural numbers {\Delta} \geq 2 and D \geq 1, find the maximum number N({\Delta},D) of vertices in a graph of maximum degree {\Delta} and diameter D. In this context, the…

Following recent work by Koll\'{a}r and Sarnak, we study gaps in the spectra of large connected cubic and quartic graphs with minimum spectral gap. We focus on two sequences of graphs, denoted $\Delta_n$ and $\Gamma_n$ which are more…

An edge-girth-regular graph $egr(n,k,g,\lambda)$ is a $k-$regular graph of order $n$, girth $g$ and with the property that each of its edges is contained in exactly $\lambda$ distinct $g-$cycles. We present new families of edge-girth…

Amply regular graphs are graphs with local distance-regularity constraints. In this paper, we prove a weaker version of a conjecture proposed by Qiao, Park, and Koolen on diameter bounds of amply regular graphs and make new progress on…

We complete the analysis of the extremal eigenvalues of the the adjacency matrix $A$ of the Erd\H{o}s-R\'enyi graph $G(N,d/N)$ in the critical regime $d \asymp \log N$ of the transition uncovered in [arXiv:1704.02953,arXiv:1704.02945],…

We study the maximum dimension $d=d(n,p)$ for which an Erd\H{o}s-R\'enyi $G(n,p)$ random graph is $d$-rigid. Our main results reveal two different regimes of rigidity in $G(n,p)$ separated at $p_c=C_*\log n/n,~C_*=2/(1-\log 2)$ -- the point…

Let $G$ be a connected graph on $n$ vertices with diameter $d$. It is known that if $2\le d\le n-2$, there are at most $n-d$ Laplacian eigenvalues in the interval $[n-d+2, n]$. In this paper, we show that if $1\le d\le n-3$, there are at…

We use two variational techniques to prove upper bounds for sums of the lowest several eigenvalues of matrices associated with finite, simple, combinatorial graphs. These include estimates for the adjacency matrix of a graph and for both…

We prove that, for any $t\ge 3$, there exists a constant $c=c(t)>0$ such that any $d$-regular $n$-vertex graph with the second largest eigenvalue in absolute value~$\lambda$ satisfying $\lambda\le c d^{t-1}/n^{t-2}$ contains vertex-disjoint…

A well known upper bound for the spectral radius of a graph, due to Hong, is that $\mu_1^2 \le 2m - n + 1$. It is conjectured that for connected graphs $n - 1 \le s^+ \le 2m - n + 1$, where $s^+$ denotes the sum of the squares of the…

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…

Jord\'an and Tanigawa recently introduced the $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G$. This is a quantitative measure of the $d$-dimensional rigidity of $G$ which generalizes the well-studied notion of spectral…

We investigate graph based secret sharing schemes and its information ratio, also called complexity, measuring the maximal amount of information the vertices has to store. It was conjectured that in large girth graphs, where the interaction…

In this paper, we study the higher Steklov eigenvalues of graphs on surfaces. We obtain the upper bound of higher Steklov eigenvalues of a finite graph $G$ with boundary $B$ and genus $g$ by using metrical deformation via probability flows.…

This paper systematically studies the behavior of the leading eigenvectors for independent edge undirected random graphs generated from a general latent position model whose link function is possibly infinite rank and also possibly…

We generalize the classical sharp bounds for the largest eigenvalue of the normalized Laplace operator, $\frac{N}{N-1}\leq \lambda_N\leq 2$, to the case of chemical hypergraphs.