In designing communication networks (graphs), number of spanning trees plays a vital and significant role, as the more quality and perfect the network, the greater the number of trees spanning this network, and this leads to greater possibilities for the connection between two vertices, and this ensures good rigidity and resistance.
In this work, we derive an obvious formulas for the number of spanning trees ( complexity) graphs generated by duplicating edge by a vertex of the path, cycle and wheel graphs. Also clear expressions of complexity of duplicating a vertex by an edge of path and cycle graphs. The eigenvalues of the Laplacian matrix of a graph are known as the Laplacian spectrum.
Furthermore by using the spectrum of Laplacian matrix we deduce an evident formula of the complexity of the shadow graph of the path graph, cycle graph and complete graphs. These explicit formulas have been found out by utilizing techniques from linear algebra, matrix theory, and orthogonal polynomials.