A graph theoretical analysis of the number of edges in k-dense graphs

Due to the increasing discovery and implementation of networks within all disciplines of life, thestudy of subgraph connectivity has become increasingly important. Motivated by the idea of community (or sub-graph) detection within a network/graph, we focused on finding characterizations of k-dense communities. For each edge uv (element of) E(G), the edge multiplicity of uv in G is given by mG(uv) = |NG(u) (intersection) NG(v)|: For an integer k with k ≥ 2, a k-dense community of a graph G, denoted by DCk(G), is a maximal connected subgraph of G induced by the vertex set VDCk(G) = {v (element of) V (G) : (exists)u (element of) V (G) such that uv (element of) E(G) and mDCk(G)(uv) ≥ k-2}. In this research, we characterize which graphs are k-dense but not (k + 1)-dense for some values of k and study the minimum and maximum number of edges such graphs can have. A better understanding of k-dense sub-graphs (or communities) helps in the study of the connectivity of large complex graphs (or networks) in the real world.