# Edge Regular Graphs

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## Abstract – Edge Regular Graphs

An edge-regular graph is a regular graph in which, for some $\lambda$, any two adjacent vertices have exactly $\lambda$ common neighbors. The below presentation is about the existence and structure of edge-regular graphs with $\lambda = 1$ and about edge-regular graphs with $\lambda > 1$ which have local neighborhood structure analogous to that of the edge-regular graphs with $\lambda = 1$.

## Abstract – Regular Clique Assembly

A regular clique assembly (RCA) is a regular graph G with positive degree satisfying: (1) every maximal clique in G is maximum and (2) each edge in G belongs to exactly one maximum clique. The RCA’s with a clique number of two are simply the regular triangle-free graphs, and the RCA’s with a clique number of three are the regular graphs such that every pair of adjacent vertices have exactly one common neighbor. There are RCA’s of every clique number greater than one, but as the clique number goes up, they are harder to find. We prove that every RCA is isomorphic to the clique graph of its clique graph. Moreover, every triangle-free regular graph is the clique graph of its line graph.