# Sesqui-regular graphs with fixed smallest eigenvalue

## TL;DR

This paper investigates sesqui-regular graphs, a generalization of strongly regular graphs, establishing bounds on their parameters based on eigenvalues and large degree conditions, extending previous algebraic graph theory results.

## Contribution

It introduces sesqui-regular graphs and proves new bounds on their parameters related to eigenvalues and degree, generalizing known results for strongly regular graphs.

## Key findings

- For large degree k, either c is bounded by λ^2(λ-1) or v-k-1 is bounded by a quadratic expression.
- The paper extends bounds known for strongly regular graphs to a broader class of sesqui-regular graphs.
- It provides algebraic conditions linking eigenvalues, degree, and parameters of sesqui-regular graphs.

## Abstract

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.