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

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