Some new 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 τ and θ such that | S i ∩ S j | = τ for any two adjacent vertices i and j , and | S i ∩ S j | = θ 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 . Let G be a walk regular graph of order 4 k + 1 and degree 2 k which is cospectral to its complement G ¯ . Let H i be switching equivalent to G i with respect to S i ⊆ V ( G i ) . We here prove that G is strongly regular if and only if Δ ( G i ) = Δ ( H i ) for i = 1 , 2 , … , 4 k + 1 , where Δ ( G ) is the number of triangles of a graph G .