  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.
