Question

In graph theory, an edge regular graph is defined as follows. Let G = (V,E) be a regular graph with v vertices and degree k. G is said to be edge regular if there is also integer λ such that:

Every two adjacent vertices have λ common neighbors.

A graph of this kind is sometimes said to be an er(v,k,λ).

I want know about eigenvalues of edge regular graph, how can we find eigenvalue of this graph?

Answer 1

For the special case when every two non-adjacent vertices have exactly c1 or c2 neighbours, there are some eigenvalue bounds in my paper:

http://www.tandfonline.com/doi/abs/10.1080/03081080600867210

Answer 2

In the event that every non adjacent pair has a fixed number $\mu$ of common neighbors this is a strongly regular graph srg$(v,k,\lambda,\mu)$ for which the 3 distinct eigenvalues and their multiplicities are known: http://en.wikipedia.org/wiki/Strongly_regular_graph (in this case the graph is diameter 2 or a disjoint union of isomorphic complete graphs.) An erg with k=2 and $\lambda=0$ is a disjoint union of cycles (none of length 3) and could have a wide range of eigenvalues. If you want connected then with v=12 k=3 and $\lambda=0$ you could have a wide variety of graphs obtained by connecting each point to three others without creating triangles (such as the skeleton of a dodecahedron, or of a hexagonal prism) all would have 3 as the unique largest eigenvalue but the rest of the spectrum could be many things.