Some Results on Walk Regular Graphs Which Are Cospectral to Its Complement

We say that a regular graph G of order n and degree r >= 1 (which is not the complete graph) is strongly regular if there exist non-negative integers tau and theta such that vertical bar S-i boolean AND S-j vertical bar = tau for any two adjacent vertices i and j, and vertical bar S-i boolean AND S-j vertical bar = theta for any two distinct non-adjacent vertices i and j, where S-k denotes the neighborhood of the vertex k. We say that a graph G of order n is walk regular if and only if its vertex deleted subgraphs G(i) = G \ i are cospectral for i = 1, 2, ... , n. We here establish necessary and sufficient conditions under which a walk regular graph G which is cospectral to its complement (G) over bar is strongly regular.