Complexity of some of Pyramid Graphs Created from a Gear Graph

In mathematics, one always aims to obtain new frameworks from specific ones. This also stratified to the regality of graphs, where one can produce numerous new graphs from a specific set of graphs. In this work we define some classes of pyramid graphs created by a gear graph and we derive straightforward formulas of the complexity of these graphs, using linear algebra matrix analysis techniques and employing knowledges of Chebyshev polynomials.


Introduction
The graph theory is a theory gathering computer science and mathematics, permitting to solve considerable problems in several fields ( telecom, social network, molecules, computer network, genetics,…) by designing them by graphs and facilitate them by idealistic cases such as the spanning trees See [ 1-10 ].
A spanning tree of a finite connected graph G is a maximal subset of the edges that contains no cycle, equivalently a minimal subset of the edges that connects all the vertices.The history of enumerating number of spanning trees () G

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of a graph G dates back into year 1842 in which the physicist of Kirchhoff [11] gave 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 of a graph G .That is if 1 if and is adjacent to ,  where i a denotes the degree of vertex .i This method allows beneficial results for a graph comprising a small number of vertices, but it will be infeasible for large graph.There are one more methods for calculating () G  ., respectively, and I is the kk  identity matrix.The characteristic of this formula is to express () G  straightway as a determinant rather than in terms of cofactors as in Kirchhoff theorem or eigenvalues as in Kelmans and Chelnokov formula.

Chebyshev Polynomial
In this part we insert some relations regarding Chebyshev polynomials of the first and second types which we use it in our calculations.
Furthermore, we rendering that the Chebyshev polynomials of the first type are defined by 1 sin ( cos ) ( ) ( ) sin (cos ) It is easily confirmed that where the conformity is valid for all complex x (except at 1 x = , where the function can be taken as the limit).
The definition of () Ux easily yields its zeros and it can therefore be confirmed that One further notes that From Eqs. ( 6) and ( 7) , we have: Finally, straightforward manipulation of the above formula gets the following formula (9), which is highly beneficial to us latter: Moreover, one can see that Now we introduce the following important two Lemmas.
Lemma 2.1 [15] Let () . Suppose that A and D are nonsingular matrices, then: This Lemma give a type of symmetry for some matrices which simplify our calculations of the complexity of graphs studied in this paper.Fig. 1 The pyramid graph (3)

Preprints
Using Lemma 2.2, yields denote the eignvalues of the matrix L of a graph G with n vertices.Kelmans and Chelnokov[12] have derived that

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Many works have conceived techniques to derive the number of spanning tree of a graph.Now, we give the following Lemma: adjacency and degree matrices of c G , the complement of G

u
edges and m sets of vertices, say, See Fig.2.