Complexity of Products of Some Complete and Complete Bipartite Graphs

The number of spanning trees in graphs (networks) is an important invariant; it is also an important measure of reliability of a network. In this paper, we derive simple formulas of the complexity, number of spanning trees, of products of some complete and completebipartitegraphssuchascartesianproduct,normalproduct,compositionproduct,tensorproduct,andsymmetricproduct, usinglinearalgebraandmatrixanalysistechniques


Introduction
In this work we deal with simple and finite undirected graphs  = (, ), where  is the vertex set and  is the edge set.For a graph , a spanning tree in  is a tree which has the same vertex set as .The number of spanning trees in , also called the complexity of the graph, denoted by (), is a wellstudied quantity (for long time).A classical result of Kirchhoff [1], can be used to determine the number of spanning trees for  = (, ).Let  = {V 1 , V 2 , . . ., V  }; then the Kirchhoff matrix  defined as ×, characteristic matrix,  = −, where  is the diagonal matrix whose elements are the degrees of the vertices of .While  is the adjacency matrix of ,  = [  ] is defined as follows: (i)   = −1V  and V  are adjacent and  ̸ = , (ii)   equals the degree of vertex V  if  = , (iii)   = 0 otherwise.

Number of Spanning Trees of Normal Product of Graphs
The normal product, or the strong product,  1 ∘  2 , is the simple graph with ( Theorem 7.For ,  ≥ 1, we have Proof.Applying Lemma 1, we have In particular, Theorem 8.For ,  ≥ 1, we have Proof.Applying Lemma 1, we have Using Lemma 2, we have In paricular,

Number of Spanning Trees of Composition Product of Graphs
The composition, or lexicographic product,  1 [ 2 ], is the simple graph with  1 ×  2 as the vertex set in which the vertices ( 1 ,  2 ) and Theorem 9.For ,  ≥ 1, we have Proof.Applying Lemma 1, we have ) ) Thus, ) ) ) ) ) ) Using Lemma 2, we have In particular,

Number of Spanning Trees of Symmetric Product of Graphs
The symmetric product,  1 ⊕  2 , is the simple graph with ( 1 ∘  2 ) =  1 ×  2 , where ( 1 ,  2 ) and (V 1 , V 2 ) are adjacent in  1 ⊕ 2 if and only if either  1 is adjacent to V 1 in  1 and  2 is not adjacent to V 2 in  2 , or  1 is not adjacent to V 1 in  1 and  2 is adjacent to V 2 in  2 [13].

Conclusion
The number of spanning trees () in graphs (networks) is an important invariant.The evaluation of this number is not only interesting from a mathematical (computational) perspective but is also an important measure of reliability of a network and designing electrical circuits.Some computationally hard problems such as the travelling salesman problem can be solved approximately by using spanning trees.Due to the high dependence of the network design and reliability on the graph theory, we introduced the above important theorems and lemmas and their proofs.