Related papers: Regular Graphs with Minimum Spectral Gap

We give an upper bound on the maximal eigenvalue of the adjacency matrix of a connected graph in terms of its maximum degree, diameter and order. This bound is best possible up to a constant factor and improves prevoius results of…

It was conjectured by Alon and proved by Friedman that a random $d$-regular graph has nearly the largest possible spectral gap, more precisely, the largest absolute value of the non-trivial eigenvalues of its adjacency matrix is at most…

A $d$-regular graph on $n$ nodes has at most $T_{\max} = \frac{n}{3} \tbinom{d}{2}$ triangles. We compute the leading asymptotics of the probability that a large random $d$-regular graph has at least $c \cdot T_{\max}$ triangles, and…

We give lower bounds on the size and total size of clique partitions of a graph in terms of its spectral radius and minimum degree, and derive a spectral upper bound for the maximum number of edge-disjoint $t$-cliques. The extremal graphs…

Let G be a simple graph without isolated vertices. For a vertex i in G, the degree d_i is the number of vertices adjacent to i and the average 2-degree m_i is the mean of the degrees of the vertices which are adjacent to i. The sequence of…

A coupling of random walkers on the same finite graph, who take turns sequentially, is said to be an avoidance coupling if the walkers never collide. Previous studies of these processes have focused almost exclusively on complete graphs, in…

A dissociation set in a graph is a subset of vertices which induces a subgraph with maximum degree at most one. The dissociation number of a graph is the maximum cardinality of its dissociation sets. In this paper, we consider the…

We describe a new method to remove short cycles on regular graphs while maintaining spectral bounds (the nontrivial eigenvalues of the adjacency matrix), as long as the graphs have certain combinatorial properties. These combinatorial…

Asymptotic properties of random regular graphs are object of extensive study in mathematics. In this note we argue, based on theory of spin glasses, that in random regular graphs the maximum cut size asymptotically equals the number of…

We study Aldous' conjecture that the spectral gap of the interchange process on a weighted undirected graph equals the spectral gap of the random walk on this graph. We present a conjecture in the form of an inequality, and prove that this…

Spectral embedding of graphs uses the top k non-trivial eigenvectors of the random walk matrix to embed the graph into R^k. The primary use of this embedding has been for practical spectral clustering algorithms [SM00,NJW02]. Recently,…

In this paper, we determine the graphs which have the minimal spectral radius among all the connected graphs of order $n$ and the independence number $\lceil\frac{n}{2}\rceil-1.$

Let $G$ be a graph with adjacency matrix $A(G)$ and degree diagonal matrix $D (G)$. In 2017, Nikiforov defined the matrix $A_\alpha(G) = \alpha D(G) + (1-\alpha)A(G)$ for any real $\alpha\in[0,1]$. The largest eigenvalue of $A_\alpha(G)$ is…

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…

We investigate the average hitting times of simple random walks on the $k$-th power graph $C_N^k$ of the cycle graph $C_N$. First, we show that the average hitting times are characterized by a difference equation corresponding to the graph…

We give a bound for the graph energy with given maximal degree in terms of the second and fourth moments of a graph. In the case in which the graph is $d$-regular we obtain the bound that is given in Van Dam, E. et al. (2014). through…

We prove that a simple random walk on quasi-transitive graphs with the volume growth being faster than any polynomial of degree 4 has a.s. infinitely many cut times, and hence infinitely many cutpoints. This confirms a conjecture raised by…

A graph is said to be walk-regular if, for each $\ell \geq 1$, every vertex is contained in the same number of closed walks of length $\ell$. We construct a $24$-vertex graph $H_4$ that is not walk-regular yet has maximized walk entropy,…

The Erd\H{o}s-Hajnal conjecture says that, for every graph $H$, there exists $c>0$ such that every $H$-free graph on $n$ vertices has a clique or stable set of size at least $n^c$. In this paper we are concerned with the case when $H$ is a…

The circumference denoted by $c(G)$ of a graph $G$ is the length of its longest cycle. Let $\delta(G)$ and $\omega(G)$ denote the minimum degree and the clique number of a graph $G$, respectively. In [\emph{Electron. J. Combin.} 31(4)(2024)…