IJPAM: Volume 81, No. 3 (2012)

INDEPENDENT AND VERTEX COVERING NUMBER
ON TENSOR PRODUCT OF FAN GRAPH

Siriluk Intaja$^1$, Thanin Sitthiwirattham$^2$
$^{1,2}$Department of Mathematics
Faculty of Applied Science
King Mongkut's University of Technology North Bangkok
Bangkok, 10800, THAILAND
$^2$Centre of Excellence in Mathematics, CHE
Sri Ayutthaya Road, Bangkok, 10400, THAILAND


Abstract. Let $\alpha(G)$ and $\beta(G)$ be the independent number and vertex covering number of $G$, respectively. The Tensor product $G_1 \otimes G_2$ of graph of $G_1$ and $G_2$ has vertex set $V(G_1 \otimes G_2) = V(G_1) \times V(G_2)$ and edge set $E(G_1 \otimes G_2)= \{(u_1 v_1)(u_2 v_2)\mid u_1 u_2 \in E(G_1)$ and $v_1 v_2 \in E(G_2)\}$. In this paper, let $G$ is a simple graph with order $p$, we prove that, $\alpha(F_{m,n}\otimes
G)=\max\{(m+n)\alpha(G),p\max\{m,\lceil\frac{n}{2}\rceil\}\}$ and $\beta(F_{m,n}\otimes(G)=\min\{(m+n)\beta(G),p\min\{m+\lfloor\frac{n}{2}\rfloor,n\}\}$.

Received: August 29, 2012

AMS Subject Classification: 05C69, 05C70, 05C76

Key Words and Phrases: tensor product, independent number, vertex covering number

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 81
Issue: 3