Vulnerability parameters of tensor product of complete equipartite graphs

Tyt. z nagłówka.Bibliogr. s. 749.Let G1 and G2 be two simple graphs. The tensor product of G1 and G2, denoted by G1 × G2, has vertex set V (G1 × G2) = V (G1) × V (G2) and edge set E(G1 × G2) ={(u1, v1)(u2, v2) : u1u2 ∈ E(G1) and v1v2 ∈ E(G2)}. In this paper, we determine vulnerability parameters such as toughness, scattering number, integrity and tenacity of the tensor product of the graphs Kr(s) × Km(n) for r ≥ 3,m ≥ 3, s ≥ 1 and n ≥ 1, where Kr(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.Dostępny również w formie drukowanej.KEYWORDS: fault tolerance, tensor product, vulnerability parameters

Decomposition of the Tensor Product of Complete Graphs into Cycles of Lengths 3 and 6

By a $\{C_3^α, C_3^β\}$-decomposition of a graph $G$, we mean a partition of the edge set of $G$ into $\alpha$ cycles of length 3 and $\beta$ cycles of length 6. In this paper, necessary and sufficient conditions for the existence of a $\{C_3^α, C_3^β\}$-decomposition of $(K_m × K_n)(\lambda)$, where $×$ denotes the tensor product of graphs and $\lambda$ is the multiplicity of the edges, is obtained. In fact, we prove that for $\lambda ≥ 1, m, n ≥ 3$ and $(m, n) ≠ (3, 3)$, a $\{C_3^α, C_3^β\}$-decomposition of $(Km × Kn)(\lambda)$ exists if and only if $\lambda(m − 1)(n − 1) ≡ 0 (mod 2)$ and $3\alpha+6\beta=\frac{\lambda m(m-1)n(n-1)}{2}$

C7-Decompositions of the Tensor Product of Complete Graphs

In this paper we consider a decomposition of Km × Kn, where × denotes the tensor product of graphs, into cycles of length seven. We prove that for m, n ≥ 3, cycles of length seven decompose the graph Km × Kn if and only if (1) either m or n is odd and (2) 14 | m(m − 1)n(n − 1). The results of this paper together with the results of [Cp-Decompositions of some regular graphs, Discrete Math. 306 (2006) 429–451] and [C5-Decompositions of the tensor product of complete graphs, Australasian J. Combinatorics 37 (2007) 285–293], give necessary and sufficient conditions for the existence of a p-cycle decomposition, where p ≥ 5 is a prime number, of the graph Km × Kn

Harary Index of Product Graphs

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, the exact formulae for the Harary indices of tensor product $G × K_{m_0,m_1,...,m_{r−1}}$ and the strong product $G⊠K_{m_0,m_1,...,m_{r−1}}$, where $K_{m_0,m_1,...,m_{r−1}}$ is the complete multipartite graph with partite sets of sizes $m_0,m_1, . . .,m_{r−1}$ are obtained. Also upper bounds for the Harary indices of tensor and strong products of graphs are estabilished. Finally, the exact formula for the Harary index of the wreath product $G ○ G′$ is obtained

Vulnerability parameters of tensor product of complete equipartite graphs

Let G1 and G2 be two simple graphs. The tensor product of G1 and G2, denoted by G1 × G2, has vertex set V (G1 × G2) = V (G1) × V (G2) and edge set E(G1 × G2) ={(u1, v1)(u2, v2) : u1u2 ∈ E(G1) and v1v2 ∈ E(G2)}. In this paper, we determine vulnerability parameters such as toughness, scattering number, integrity and tenacity of the tensor product of the graphs Kr(s) × Km(n) for r ≥ 3,m ≥ 3, s ≥ 1 and n ≥ 1, where Kr(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

Hamilton cycle decompositions of the tensor products of complete bipartite graphs and complete multipartite graphs

AbstractIn this paper, it is shown that the tensor product of the complete bipartite graph, Kr,r,r≥2, and the regular complete multipartite graph, Km∗K¯n,m≥3, is Hamilton cycle decomposable

C₇-Decompositions of the Tensor Product of Complete Graphs

In this paper we consider a decomposition of Km × Kn, where × denotes the tensor product of graphs, into cycles of length seven. We prove that for m, n ≥ 3, cycles of length seven decompose the graph Km × Kn if and only if (1) either m or n is odd and (2) 14 | m(m − 1)n(n − 1). The results of this paper together with the results of [Cp-Decompositions of some regular graphs, Discrete Math. 306 (2006) 429–451] and [C5-Decompositions of the tensor product of complete graphs, Australasian J. Combinatorics 37 (2007) 285–293], give necessary and sufficient conditions for the existence of a p-cycle decomposition, where p ≥ 5 is a prime number, of the graph Km × Kn

Wiener and vertex PI indices of the strong product of graphs

The Wiener index of a connected graph G, denoted by W(G), is defined as $½ ∑_{u,v ∈ V(G)}d_G(u,v)$. Similarly, the hyper-Wiener index of a connected graph G, denoted by WW(G), is defined as $½W(G) + ¼ ∑_{u,v ∈ V(G)} d²_G(u,v)$. The vertex Padmakar-Ivan (vertex PI) index of a graph G is the sum over all edges uv of G of the number of vertices which are not equidistant from u and v. In this paper, the exact formulae for Wiener, hyper-Wiener and vertex PI indices of the strong product $G ⊠ K_{m₀,m₁,...,m_{r -1}}$, where $K_{m₀,m₁,...,m_{r -1}}$ is the complete multipartite graph with partite sets of sizes $m₀,m₁, ...,m_{r -1}$, are obtained. Also lower bounds for Wiener and hyper-Wiener indices of strong product of graphs are established