A new approach for the characteristic polynomial of a complete tripartite graph

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The case k = 3 of a complete k-partite graph is called a complete tripartite graph Tp,q,r. It is a graph that its vertices are decomposed into three disjoint sets such that, no two graph vertices within the same set are adjacent. It has recently attracted much attention due to its important in several applications, especially, in chemistry where some of the molecular orbital compounds are correspondents to the tripartite graph structure. One method of capturing graph structure is through computing of the characteristic polynomial for the matrix characterization M of a graph, which is defined as the determinant | λI – M | where I is the identity matrix and λ is the variable of the polynomial. The general technique of the characteristic polynomials evaluation of graphs with large number of vertices is considered as an extremely tiresome problem when it is based on matrix, because its computational complexity is high. In this paper, a new approach for the characteristic polynomial of a complete tripartite graph Ti,i,n−2i, for n ≥ 4, based on the adjacency matrix is introduced. It shows good efficiency because it reduces the complexity and the difficulty of computation in comparing to some well-known methods especially, for the graphs with large number of vertices.