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In this paper, we derived the explicit formula of Chebyshev polynomials of the third kind and the fourth kind by using a composita FΔ(n,k) of a generating function F(t) = ∑n>0fn tn . By multiplying (1 - t) to the composition of generating function, G(t) = 1/1-t and F(x,t) = 2xt - t², that has a composita FΔ(n,k), the coefficients of chebysshev polynomials of the third kind can be found. Moreover, by multiplying (1 + t) to the composition of generating functions G(t) = 1/1-t and F(x,t) = 2xt - t², that has a composita FΔ (n,k), the coefficients of chebyshev polynomials of the fourth kind can be obtained. From those coefficients, the explicit formula of chebyshev polynomials of the third and the fourth kinds are derived. [ABSTRACT FROM AUTHOR]

An adjacency matrix A(G) of directed graph G is an m×m matrix consisting of only entries 0 and 1, where m is the number of vertices of G. The entry aij is equal to 1 if there exists a directed edge from vertex vi to vertex vj, otherwise it is equal to 0. Let D(G) be a diagonal matrix of size m×m with each of its main diagonal entry being the degree of the corresponding vertex of directed graph G. Then the matrix L(G) = D(G) − A(G) is called the Laplacian matrix of G. Since a directed graph has two types of degrees namely indegree and outdegree, they result in directed graphs also having the both types of its Laplacian matrices. In this study, the adjacency matrix and the Laplacian matrix of cyclic directed prism graph are investigated. The general form of the coefficients of polynomial characteristic of the Adjacency matrix and Laplacian matrix, respectively is obtained by applying the row reduction method in linear algebra, whereas the general form of the eigenvalues of the polynomial characteristic of the Adjacency matrix and Laplacian matrix, respectively is obtained by factorization and substitution methods. [ABSTRACT FROM AUTHOR]

A prism graph is a graph which corresponds to the skeleton of an n-prism and therefore it is a cyclic simple graph. It is denoted Yn (n ≥ 3) where n is half the number of vertices. An n-prism graph has 2n vertices and 3n edges. In this paper, only regularly-directed cyclic prism graphs are investigated. The anti-adjacency matrix is applied as the graph representation. An anti-adjacency matrix of graph representation is a 0–1 matrix of size m × m where m is the number of vertices. The entry bij of an anti-adjacency matrix B(G) of directed graph G is 0 if there exists a directed edge from vertex vi to vertex vj and is 1 otherwise. The characteristic polynomial of the anti-adjacency matrix B(Yn) of directed cyclic prism graph Yn are obtained. The characteristic polynomial will be proved by observing the both cyclic and acyclic induced subgraphs of the directed cyclic prims graph. Furthermore, the anti-adjacency matrix of directed cyclic prism graph is found to have both real eigenvalues and complex eigenvalues which appear in conjugate pairs. [ABSTRACT FROM AUTHOR]