Multiplicative Zagreb Indices and Extremal Complexity of Line Graphs

The number of spanning trees of a graph G is called the complexity of G. It is known that the complexity of the line graph of a given graph G can be computed as the sum over all spanning trees of G of contributions which depend on various types of products of degrees of vertices of G. We interpret the contributions in terms of three types of multiplicative Zagreb indices, obtaining simple and compact expressions for the complexity of line graphs of graphs with low cyclomatic numbers. As an application, we determine the unicyclic graphs whose line graphs have the smallest and the largest complexity.