On some covering graphs of a graph
| Main Authors: | Pirzada, Shariefuddin; Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India, Ganie, Hilal A; Department of Mathematics, University of Kashmir, Srinagar, Kashmir, India, Siddique, Merajuddin; Department of Applied Mathematics, Aligarh Muslim University, Aligarh, India |
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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/166 http://www.ejgta.org/index.php/ejgta/article/view/166/pdf_22 |
Daftar Isi:
- For a graph $G$ with vertex set $V(G)=\{v_1, v_2, \dots, v_n\}$, let $S$ be the covering set of $G$ having the maximum degree over all the minimum covering sets of $G$. Let $N_S[v]=\{u\in S : uv \in E(G) \}\cup \{v\}$ be the closed neighbourhood of the vertex $v$ with respect to $S.$ We define a square matrix $A_S(G)= (a_{ij}),$ by $a_{ij}=1,$ if $\left |N_S[v_i]\cap N_S[v_j] \right| \geq 1, i\neq j$ and 0, otherwise. The graph $G^S$ associated with the matrix $A_S(G)$ is called the maximum degree minimum covering graph (MDMC-graph) of the graph $G$. In this paper, we give conditions for the graph $G^S$ to be bipartite and Hamiltonian. Also we obtain a bound for the number of edges of the graph $G^S$ in terms of the structure of $G$. Further we obtain an upper bound for covering number (independence number) of $G^S$ in terms of the covering number (independence number) of $G$.