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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 $\Gamma$ with respect to both $x$ and $y$ is equitable, and the parameters corresponding to equitable partitions are independent of the choice of $x$ and $y$. Assume $\Gamma$ is $1$-homogeneous distance-regular with intersection number $a_1>0$ and $D\geqslant 5$. Define $b=b_1/(\theta_1+1)$, where $b_1$ is the intersection number and $\theta_1$ is the second largest eigenvalue of $\Gamma$. We show that if intersection number $c_2\geqslant 2$, then $b\geqslant 1$ and one of the following (i)--(vi) holds: (i) $\Gamma$ is a regular near $2D$-gon, (ii) $\Gamma$ is a Johnson graph $J(2D,D)$, (iii) $\Gamma$ is a halved $\ell$-cube where $\ell \in \{2D,2D+1\}$, (iv) $\Gamma$ is a folded Johnson graph $\bar{J}(4D,2D)$, (v) $\Gamma$ is a folded halved $(4D)$-cube, (vi) the valency of $\Gamma$ is bounded by a function of $b$. Using this result, we characterize $1$-homogeneous graphs with classical parameters and $a_1>0$, as well as tight distance-regular graphs., Comment: 19 pages

In this paper we study when the $q$-distance matrix of a distance-regular graph has few distinct eigenvalues. We mainly concentrate on diameter 3., Comment: 15 pages

In this paper, we study the $q$-distance matrix for a distance-regular graph and show that the $q$-distance matrix of a distance-regular graph with classical parameters ($D, q, \alpha, \beta$) has exactly three distinct eigenvalues, of which one is zero. Moreover, we study distance-regular graphs whose $q$-distance matrix has exactly one positive eigenvalue., Comment: 19 pages

In this paper we show that there does not exist a strongly regular graph with parameters $(1911,270,105,27)$., Comment: 13 pages, 2 tables

Koolen et al. showed that if a graph with smallest eigenvalue at least $-3$ has large minimal valency, then it is $2$-integrable. In this paper, we will focus on the sesqui-regular graphs with smallest eigenvalue at least $-3$ and study their integrability.

In this paper we give two characterizations of the $p \times q$-grid graphs as co-edge-regular graphs with four distinct eigenvalues., Comment: 18 pages, 1 table

Let $\lambda\geq2$ be an integer. For strongly regular graphs with parameters $(v, k, a,c)$ and smallest eigenvalue $-\lambda$, Neumaier gave two bounds on $c$ by using algebraic property of strongly regular graphs. In this paper, we will study a new class of regular graphs called sesqui-regular graphs, which contains strongly regular graphs as a subclass, and prove that for a sesqui-regular graph with parameters $(v,k,c)$ and smallest eigenvalue at least $-\lambda$, if $k$ is very large, then either $c \leq \lambda^2(\lambda -1)$ or $v-k-1 \leq \frac{(\lambda-1)^2}{4} + 1$ holds.