Some characterizations of strongly regular graphs with respect to their vertex deleted subgraphs

Abstract

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 . Let i be a fixed vertex from the vertex set V ( G ) = { 1 , 2 , … , n } and let G i = G ∖︀ i be its vertex deleted subgraph. Let H i be switching equivalent to G i with respect to S i ⊆ V ( G i ) . We prove that H i is a strongly regular graph of order n − 1 and degree 2 ( r − θ ) with τ ( H i ) = r + τ − 2 θ and θ ( H i ) = r − θ if and only if n = 4 r − 2 τ − 2 θ . Otherwise, we prove that H i has exactly two main eigenvalues μ 1 and μ 2 . In this case, the main eigenvalues of H i are represented in the form μ 1 , 2 = τ − θ + r ± ( τ − θ + r ) 2 + 4 ( n − 2 r ) ( r − θ ) 2 . We also prove that G i and H i are cospectral if and only if r = 2 θ . Finally, we show that if G is a strongly regular graph of order 2 ( 2 k 2 + 2 k + 1 ) and degree k ( 2 k + 1 ) with τ = k 2 − 1 and θ = k 2 then H i is a conference graph of order ( 2 k + 1 ) 2 and degree 2 k ( k + 1 ) with τ ( H i ) = k 2 + k − 1 and θ ( H i ) = k ( k + 1 ) . However, if G is a strongly regular graph of order 4 ( 3 k 2 + 3 k + 1 ) and degree 3 k ( 2 k + 1 ) with τ = 3 k 2 − k − 2 and θ = k ( 3 k + 1 ) then H i is a strongly regular graph (non-conference) of order 3 ( 2 k + 1 ) 2 and degree 2 k ( 3 k + 2 ) with τ ( H i ) = 3 k 2 − 2 and θ ( H i ) = k ( 3 k + 2 ) .