A Graph Theoretical Analysis of the Number of Edges in k-dense Graphs

TL;DR

This paper characterizes k-dense subgraphs in networks, analyzing their edge counts and properties to better understand community structures within complex graphs.

Contribution

It provides new characterizations of k-dense graphs and explores their edge bounds, advancing understanding of community detection in network analysis.

Findings

Characterization of graphs that are k-dense but not (k+1)-dense.

Determination of minimum and maximum edges in such graphs.

Insights into the connectivity of complex networks.

Abstract

Due to the increasing discovery and implementation of networks within all disciplines of life, the study 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 $uv\in E(G)$, the {\bf edge multiplicity} of $uv$ in $G$ is given by $m_G(uv)=|N_{G}(u)\cap N_{G}(v)|.$ For an integer $k$ with $k\ge 2$, a {\bf $k$-dense community} of a graph $G$, denoted by $DC_k(G)$, is a maximal connected subgraph of $G$ induced by the vertex set $$V_{DC_k(G)} = \{v\in V(G) : \exists u\in V(G)\ {\rm such\ that\} uv\in E(G)\ {\rm and\} m_{DC_{k(G)}}(uv)\ge k-2\}.$$ 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.