Some Results on Overgraphs of a Strongly Regular Graph

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 λ 1 = r, λ 2 and λ 3 denote the distinct eigenvalues of G. Let m 1 = 1, m 2, and m 3 denote the multiplicity of r, λ 2, and λ 3, respectively. We demonstrate that the characteristic polynomial of the graph G S , which is obtained from the graph G by adding a new vertex adjacent exactly to the vertices from S ⊆ V (G), can be represented in the form \(P_{G_{S}}(\lambda ) = (\lambda -\lambda _{2})^{m_{2}-1}(\lambda -\lambda _{3})^{m_{3}-1}{\Delta }_{s,t}(\lambda )\), where $$\begin{array}{@{}rcl@{}} {\Delta}_{s,t}(\lambda) &=& \lambda^{4} - (\tau - \theta + r)\lambda^{3} - (s + (r-\theta) - (\tau - \theta)r)\lambda^{2}\\ &&+((\tau-\theta+r)s + (r-\theta)r - t)\lambda\\ &&-(\theta s^{2} + (\tau-\theta)rs - rt), \end{array} $$ s = |S|, \(t = {\sum }_{i\in S}{\sum }_{j\in S}a_{ij}\), and a i j are the entries of the ordinary adjacency matrix A = A[a i j [ of the graph G.