Subgroup sum graphs of finite abelian groups

Peter J. Cameron, R. Raveendra Prathap, T. Tamizh Chelvam*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


Let G be a finite abelian group, written additively, and H a subgroup of G. The subgroup sum graph ΓG,H is the graph with vertex set G, in which two distinct vertices x and y are joined if x+y∈H∖{0}. These graphs form a fairly large class of Cayley sum graphs. Among cases which have been considered previously are the prime sum graphs, in the case where H=pG for some prime number p. In this paper we present their structure and a detailed analysis of their properties. We also consider the simpler graph Γ+G,H, which we refer to as the extended subgroup sum graph, in which x and y are joined if x+y∈H: the subgroup sum is obtained by removing from this graph the partial matching of edges having the form {x,−x} when 2x≠0. We study perfectness, clique number and independence number, connectedness, diameter, spectrum, and domination number of these graphs and their complements. We interpret our general results in detail in the prime sum graphs.
Original languageEnglish
Article number114
Number of pages13
JournalGraphs and Combinatorics
Issue number4
Early online date2 Jul 2022
Publication statusPublished - 1 Aug 2022


  • Subgroup sum graph
  • Finite abelian group
  • Clique number
  • Chromatic number
  • Cayley graphs
  • Spectrum


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