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

We say that a regular graph GG of order nn and degree r≥1r≥1 (which is not the complete graph) is strongly regular if there exist non-negative integers ττ and θθ such that |Si∩Sj|=τ|Si∩Sj|=τ for any two adjacent vertices ii and jj, and |Si∩Sj|=θ|Si∩Sj|=θ for any two distinct non-adjacent vertices ii and jj, where SkSk denotes the neighborhood of the vertex kk. Let ii be a fixed vertex from the vertex set V(G)={1,2,…,n}V(G)={1,2,…,n} and let Gi=G∖︀iGi=G∖︀i be its vertex deleted subgraph. Let HiHi be switching equivalent to GiGi with respect to Si⊆V(Gi)Si⊆V(Gi). We prove that HiHi is a strongly regular graph of order n−1n−1 and degree 2(r−θ)2(r−θ) with τ(Hi)=r+τ−2θτ(Hi)=r+τ−2θ and θ(Hi)=r−θθ(Hi)=r−θ if and only if n=4r−2τ−2θn=4r−2τ−2θ. Otherwise, we prove that HiHi has exactly two main eigenvalues μ1μ1 and μ2μ2. In this case, the main eigenvalues of HiHi are represented in the form μ1,2=τ−θ+r±(τ−θ+r)2+4(n−2r)(r−θ)2. We also prove that GiGi and HiHi are cospectral if and only if r=2θr=2θ. Finally, we show that if GG is a strongly regular graph of order 2(2k2+2k+1)2(2k2+2k+1) and degree k(2k+1)k(2k+1) with τ=k2−1τ=k2−1 and θ=k2θ=k2 then HiHi is a conference graph of order (2k+1)2(2k+1)2 and degree 2k(k+1)2k(k+1) with τ(Hi)=k2+k−1τ(Hi)=k2+k−1 and θ(Hi)=k(k+1)θ(Hi)=k(k+1). However, if GG is a strongly regular graph of order 4(3k2+3k+1)4(3k2+3k+1) and degree 3k(2k+1)3k(2k+1) with τ=3k2−k−2τ=3k2−k−2 and θ=k(3k+1)θ=k(3k+1) then HiHi is a strongly regular graph (non-conference) of order 3(2k+1)23(2k+1)2 and degree 2k(3k+2)2k(3k+2) with τ(Hi)=3k2−2τ(Hi)=3k2−2 and θ(Hi)=k(3k+2)θ(Hi)=k(3k+2).