The transformation of edge-regular and pseudo strongly regular graphs under graph operations

TL;DR

This paper investigates how different graph operations affect edge-regular and pseudo strongly regular graphs, which are weaker variants of strongly regular graphs characterized by specific local adjacency properties.

Contribution

It provides a systematic analysis of the transformations of edge-regular and pseudo strongly regular graphs under various graph operations, expanding understanding of their structural behavior.

Findings

Identifies how graph operations preserve or alter edge-regularity.

Analyzes the impact of operations on pseudo strongly regular graphs.

Provides conditions under which these graph classes are maintained or transformed.

Abstract

The graph $G$ is said to be strongly regular with parameters $(n,k,\lambda,\mu)$ if the following conditions hold: (1) each vertex has $k$ neighbours; (2) any two adjacent vertices of $G$ have $\lambda$ common neighbours; (3) any two non-adjacent vertices of $G$ have $\mu$ common neighbours. In this paper we study two weaker notions of strongly regular graphs. A graph satisfying the conditions $(1)$ and $(2)$ is called an edge-regular graph with parameters $(n,k,\lambda)$. We call a graph satisfying the conditions $(1)$ and $(3)$ a pseudo strongly regular graph with parameters $(n,k,\mu)$. In this paper we study the impact of various graph operations on edge regular graphs and pseudo strongly regular graphs.