2020/08/11 by Arindam Biswas, Biswas, Arindam, Jyoti Prakash Saha +1
Engineering · Mathematics · #05C25 #05C50 #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Graph theory and applications #graph theory and CDMA systems #math.CO #msc:05C25 #msc:05C50
paper · pdf · doi:10.48550/arxiv.2008.04307
arxiv created 2020/08/11 · openalex publication_date 2020/08/11 · arxiv updated 2020/08/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let G be a finite group with |G|≥ 4 and S be a subset of G. Given an automorphism σ of G, the twisted Cayley graph C(G, S)σ (resp. the twisted Cayley sum graph CΣ(G, S)σ) is defined as the graph having G as its set of vertices and the adjacent vertices of a vertex g∈ G are of the form σ(gs) (resp. σ(g-1 s)) for some s∈ S. If the twisted Cayley graph C(G, S)σ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from -1 and this bound depends only on its degree, the order of σ and the vertex Cheeger constant of C(G, S)σ. Moreover, if the twisted Cayley sum graph CΣ(G, S)σ is undirected and connected, then we prove that the nontrivial spectrum of its normalised adjacency operator is bounded away from -1 and this bound depends only on its degree and the vertex Cheeger constant of CΣ(G, S)σ. We also study these twisted graphs with respect to anti-automorphisms, and obtain similar results. Further, we prove an analogous result for the Schreier graphs satisfying certain conditions.