Extreme Topologies on Bipolygonal Graphs and Dinamic Trees

Abstract

We show how properties of the sequence β_i,j, which represents the product between two Fibonacci's numbers F_i × F_j, can be used for the computation of the Merrifield-Simmons index on bipolygonal graphs and trees.

We show that the extreme values of the Merrifield-Simmons index on bipolygonal graphs are found in two consecutive columns of the table β_i,j k=i+j=1,...,n. The minimum value in β_3,k-3 and the maximum value in β_4,k-4.  On the other hand we show that i(T_n ∪ {{v_p, v}}) is minimum when v is a new leaf node, and its father v_p was also a leaf node in T_n.

Our methods does not require the explicit computation of the number of independent sets of the involved graphs. Instead, it is based on applying the edge and vertex division rules to decompose the initial graph.