The distance spectrum of the bipartite double cover of strongly regular graphs
S. Morteza Mirafzal
TL;DR
The paper addresses the distance spectrum of the bipartite double cover $B(G)$ of a strongly regular graph $G=(V,E)$ with parameters $(n,d,a,c)$. The authors derive a block-structured form for the adjacency matrix of $B(G)$ and express the distance matrix $D$ of $B(G)$ in terms of these blocks as $D=-2M+2M_4+2X+3Y+2M_5-2I_{2n}$ under suitable irreducibility and diameter assumptions, then obtain the distance spectrum of $D$ explicitly in terms of the spectrum of $G$. They distinguish the cases $a eq 0$ and $a=0$ (the latter excluding $G$ isomorphic to $K_{m,m}$), providing precise multiplicities, e.g., $Spec(D)=igl\\{(-2d+5n-2)^1,(2λ_1-2)^{m_1},(2λ_2-2)^{m_2},(-2λ_2-2)^{m_2},(-2λ_1-2)^{m_1},(2d-n-2)^1\bigr\\}$ and $Spec(D)=igl\\{(5n)^1,(4λ_1-4)^{m_1},(4λ_2-4)^{m_2},0^{n-1},(4d-n-4)^1\bigr\\}$ in the respective cases. A key takeaway is that integral strongly regular graphs yield distance-integral bipartite double covers, linking eigenvalue structure to distance properties in a closed form with concrete examples.
Abstract
A strongly regular graph with parameters $(n,d,a,c)$ is a $d$-regular graph of order $n$, in which every pair of adjacent vertices has exactly $a$ common neighbor(s) and every pair of nonadjacent vertices has exactly $c$ common neighbor(s). Let $n$ be the number of vertices of the graph $G=(V,E)$. The distance matrix $D=D(G)$ of $G$ is an $n \times n $ matrix with the rows and columns indexed by $V$ such that $D_{uv} = d_{G}(u, v)=d(u,v)$, where $d_{G}(u, v)$ is the distance between the vertices $u$ and $v$ in the graph $G$. In this paper, we are interested in determining the distance spectrum of the bipartite double cover of the family of strongly regular graphs. In other words, let $G=(V,E)$ be a strongly regular graph with parameters $(n,k,a,c)$. We show that there is a close relationship between the spectrum of $G$ and the distance spectrum of $B(G)$, where $B(G)$ is the double cover of $G$. We explicitly determine the distance spectrum of the graph $B(G)$, according to the spectrum of $G$. In fact, according to the parameters of the graph $G$.