Abstract
There are plentiful ways to duplicate a graph (network), such as splitting, shadow, mirror, and total graph. In this paper, we derive an evident formula of the complexity, a number of spanning trees, of the closed helm graph, the mirror graph of the path and cycle, the total graph of the path, the cycle, and the wheel. Furthermore, we got an explicit formula for the splitting of a special family of graphs such as path, cycle, complete graph $K_{n}$ , complete bipartite graph $K_{n,n}$ , prism, diagonal prism, and the graphs obtained from the wheel and double wheel by splitting the vertices on their rim. Finally, an obvious formula for the complexity of $k-$ shadow graph for some graphs such as the path, the cycle, the wheel, the complete graph, and the fan graph $F_{n}$ has been obtained. These formulas have been discovered by employing techniques from linear algebra, orthogonal polynomials, and matrix theory.
Highlights
A graph is the abstract illustration of the network
The splitting graph S ( ) of graph is obtained by adding a new vertex x corresponding to each vertex x of such that N (x) = N (x )
The k−splitting graph Sk ( ) of graph is obtained by adding to each vertex x of new k vertices, say x1, x2, · · ·, xk such that xi, 1 ≤ i ≤ k is adjacent to each vertex that is adjacent to x in, [2]
Summary
A graph is the abstract illustration of the network. Let be a finite, simple and connected network (graph), plentiful new graphs can be generated from a given pair of graphs by using graph operation [1]. By the complexity τ ( ) of the graph we mean the number of its spanning trees. For an infinite family of graphs n, n ∈ N, one can introduce complexity function τ ( ) = τ (n) i.e., we can get, obvious formulas that make it quite easier to enumerate and find out the number of corresponding spanning trees, in particular when these numbers are very large. Let λ1 ≥ λ2 ≥ · · · ≥ λn(= 0) denote the eigenvalues of H matrix, while the involved calculations are complicated and this method is not valid for calculating the number of spanning trees for larger graphs. A lot of work has designed techniques to enumerate the number of spanning tree of a graph and derive closed formulas have been published for some families of graphs that can be found in [7]–[13]
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