A graph theoretical analysis of the number of edges in k-dense graphs
| Main Authors: | Eroh, Linda; Department of Mathematics, University of Wisconsin-Oskhosh, Oshkosh, WI, USA, Escuadro, Henry; Department of Mathematics, Juniata College, Huntingdon, PA, USA, Gera, Ralucca; Department of Applied Mathematics, Naval Postgraduate School, Monterey, CA, USA, Prahlow, Samuel; Department of Mathematics and Statistics, Valparaiso University, Valparaiso, IN, USA, Schmitt, Karl; Department of Mathematics and Statistics, Department of Computer and Information Sciences, Valparaiso University, Valparaiso, IN, USA |
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| Format: | Article info application/pdf eJournal |
| Bahasa: | eng |
| Terbitan: |
GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB
, 2016
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| Subjects: | |
| Online Access: |
http://www.ejgta.org/index.php/ejgta/article/view/113 http://www.ejgta.org/index.php/ejgta/article/view/113/pdf_14 |
Daftar Isi:
- 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 subgraph) 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 $\boldsymbol{k}$-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.