Related papers: A characterization of Johnson and Hamming graphs a…

In this paper, we classify distance regular graphs such that all of its second largest local eigenvalues are at most one. Also we discuss the consequences for the smallest eigenvalue of a distance-regular graph. These extend a result by the…

We prove that a distance-regular graph with a dominant distance is a spectral expander. The key ingredient of the proof is a new inequality on the intersection numbers. We use the spectral gap bound to study the structure of the…

The motion of a graph is the minimal degree of its full automorphism group. Babai conjectured that the motion of a primitive distance-regular graph on $n$ vertices of diameter greater than two is at least $n/C$ for some universal constant…

For a distance-regular graph with second largest eigenvalue (resp. smallest eigenvalue) \mu1 (resp. \muD) we show that (\mu1+1)(\muD+1)<= -b1 holds, where equality only holds when the diameter equals two. Using this inequality we study…

Let $\Gamma$ be a graph with diameter at least two. Then $\Gamma$ is said to be $1$-homogeneous (in the sense of Nomura) whenever for every pair of adjacent vertices $x$ and $y$ in $\Gamma$, the distance partition of the vertex set of…

In 2018, by Ramsey and Hoffman theory, Koolen, Yang, and Yang presented a structural result on graphs with smallest eigenvalue at least $-3$ and large minimum degree. In this study, we depart from the conventional use of Ramsey theory and…

The minimal degree of a permutation group $G$ is the minimum number of points not fixed by non-identity elements of $G$. Lower bounds on the minimal degree have strong structural consequences on $G$. In 2014 Babai proved that the…

We prove a conjecture by Van Dam and Sotirov on the smallest eigenvalue of (distance-$j$) Hamming graphs and a conjecture by Karloff on the smallest eigenvalue of (distance-$j$) Johnson graphs. More generally, we study the smallest…

In 2010, Koolen and Bang proposed the following conjecture: For a fixed integer $m \geq 2$, any geometric distance-regular graph with smallest eigenvalue $-m$, diameter $D \geq 3$ and $c_2 \geq 2$ is either a Johnson graph, a Grassmann…

We study the minimum number of distinct eigenvalues over a collection of matrices associated with a graph. Lower bounds are derived based on the existence or non-existence of certain cycle(s) in a graph. A key result proves that every…

In this paper we give a new characterization of the dual polar graphs, extending the work of Brouwer and Wilbrink on regular near polygons. Also as a consequence of our characterization we confirm a conjecture of the authors on…

In 2017, Qiao and Koolen showed that for any fixed integer $D\geq 3$, there are only finitely many such graphs with $\theta_{\min}\leq -\alpha k$, where $0<\alpha<1$ is any fixed number. In this paper, we will study non-bipartite…

In this paper, we classify non-geometric distance-regular graphs of diameter at least $3$ with smallest eigenvalue at least $-3$. This is progress towards what is hoped to be an eventual complete classification of distance-regular graphs…

We give a survey on graphs with fixed smallest eigenvalue, especially on graphs with large minimal valency and also on graphs with good structures. Our survey mainly consists of the following two parts: (i) Hoffman graphs, the basic theory…

Let $\Gamma$ denote a distance-regular graph with diameter $D\geq 3$. Juri\v{s}i\'c and Vidali conjectured that if $\Gamma$ is tight with classical parameters $(D,b,\alpha,\beta)$, $b\geq 2$, then $\Gamma$ is not locally the block graph of…

The Cheeger constant of a graph is the smallest possible ratio between the size of a subgraph and the size of its boundary. It is well known that this constant must be at least $\frac{\lambda_1}{2}$, where $\lambda_1$ is the smallest…

Let $G$ denote a near-polygon distance-regular graph with diameter $d\geq 3$, valency $k$ and intersection numbers $a_1>0$, $c_2>1$. Let $\theta_1$ denote the second largest eigenvalue for the adjacency matrix of $G$. We show $\theta_1$ is…

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…

We classify the distance-regular Cayley graphs with least eigenvalue $-2$ and diameter at most three. Besides sporadic examples, these comprise of the lattice graphs, certain triangular graphs, and line graphs of incidence graphs of certain…

We study regular graphs whose distance-$2$ graph or distance-$1$-or-$2$ graph is strongly regular. We provide a characterization of such graphs $\Gamma$ (among regular graphs with few distinct eigenvalues) in terms of the spectrum and the…