Inverse problem for sigma index

Abstract

If G is a (molecular) graph and d(v), the degree of its vertex u, then its sigma index is defined as sigma(G) = Sigma(d(u) - d(v))(2), with summation going over all pairs of adjacent vertices. Some basic properties of sigma(G) are established. The inverse problem for topological indices is about the existence of a graph having its index value equal to a given non-negative integer. We study the problem for the sigma index and first show that sigma(G) is an even integer. Then we construct graph classes in which sigma(G) covers all positive even integers. We also study the inverse problem for acyclic, unicyclic, and bicyclic graphs.