Abstract
The methods of measuring the complexity (spanning trees) in a finite graph, a problem related to various areas of mathematics and physics, have been inspected by many mathematicians and physicists. In this work, we defined some classes of pyramid graphs created by a gear graph then we developed the Kirchhoff's matrix tree theorem method to produce explicit formulas for the complexity of these graphs, using linear algebra, matrix analysis techniques, and employing knowledge of Chebyshev polynomials. Finally, we gave some numerical results for the number of spanning trees of the studied graphs.
Highlights
IntroductionThe graph theory is a theory that combines computer science and mathematics, which can solve considerable problems in several fields (telecom, social network, molecules, computer network, genetics, etc.) by designing graphs and facilitating them through idealistic cases such as the spanning trees, see [1,2,3,4,5,6,7,8,9,10]
The graph theory is a theory that combines computer science and mathematics, which can solve considerable problems in several fields by designing graphs and facilitating them through idealistic cases such as the spanning trees, see [1,2,3,4,5,6,7,8,9,10].A spanning tree of a finite connected graph G is a maximal subset of the edges that contains no cycle, or equivalently a minimal subset of the edges that connects all the vertices
Kirchhoff [11] offered the matrix tree theorem established on the determinants of a certain matrix gained from the Laplacian matrix L defined by the difference between the degree matrix D and adjacency matrix A, where D is a diagonal matrix, D = dig (d1, d2, . . . , dn ) corresponding to a graph
Summary
The graph theory is a theory that combines computer science and mathematics, which can solve considerable problems in several fields (telecom, social network, molecules, computer network, genetics, etc.) by designing graphs and facilitating them through idealistic cases such as the spanning trees, see [1,2,3,4,5,6,7,8,9,10]. For any graph G, the complexity τ ( G ) of G is equal to τ ( G ) = τ ( G − e) + τ ( G/e), where e is any edge of G, and where G − e is the deletion of e from G, and G/e is the contraction of e in G This gives a recursive method to calculate the complexity of a graph [13,14]. Τ ( G ) = k12 det (k I − D c + Ac ) where Ac and D c are the adjacency and degree matrices of G c , the complement of G, respectively, and I is the k × k identity matrix The characteristic of this formula is to express τ ( G ) straightway as a determinant rather than in terms of cofactors as in Kirchhoff theorem or eigenvalues as in Kelmans and Chelnokov formula
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