On The Number Of Spanning Trees And Critical Groups Of Some Graphs

The number of spanning trees as an invariant of the graph has been studied widespreadly. The critcal groups of a connected graph is a finite abelian groups whose order is equal to the number of spanning trees of the graph. In this paper, we calculated the number of spanning trees of some graphs and the critical groups of those graphs are determined. The main results of this thesis are the following:1.We calculate the number of spanning trees of Some planar graphs such as ladders, fans, wheels, A_n, B_n, D_n by means of their duals using matrix-tree theorem.2.We determined the structure of critical groups of such as ladders, fans, wheels, A_n, B_n, D_n, X_n whose critical groups is the direct sum of few cyclice groups.3.The critical groups for some graphs with large number of edges such as K_b-K_1,m, K_n-K_m, K_n-mK₂,K_n.n- nK₂ are determined completely. The critical groups of these graphs are the direct sum of more cyclice groups.