Opuscula Math. 33, no. 4 (), 741-750
http://dx.doi.org/10.7494/OpMath.2013.33.4.741
Opuscula Mathematica

# Vulnerability parameters of tensor product of complete equipartite graphs

Abstract. Let $$G_{1}$$ and $$G_{2}$$ be two simple graphs. The tensor product of $$G_{1}$$ and $$G_{2}$$, denoted by $$G_{1}\times G_{2}$$, has vertex set $$V(G_{1}\times G_{2})=V(G_{1})\times V(G_{2})$$ and edge set $$E(G_{1}\times G_{2})=\{(u_{1},v_{1})(u_{2},v_{2}):u_{1}u_{2}\in E(G_{1})$$ and $$v_{1}v_{2}\in E(G_{2})\}$$. In this paper, we determine vulnerability parameters such as toughness, scattering number, integrity and tenacity of the tensor product of the graphs $$K_{r(s)}\times K_{m(n)}$$ for $$r\geq 3, m\geq 3, s\geq 1$$ and $$n\geq 1,$$ where $$K_{r(s)}$$ denotes the complete $$r$$-partite graph in which each part has $$s$$ vertices. Using the results obtained here the theorems proved in [Aygul Mamut, Elkin Vumar, Vertex Vulnerability Parameters of Kronecker Products of Complete Graphs, Information Processing Letters 106 (2008), 258-262] are obtained as corollaries.
Keywords: fault tolerance, tensor product, vulnerability parameters.
Mathematics Subject Classification: 05C76, 05C40.