On the product of Sombor and modified Sombor indices

Abstract

The Sombor index (SO) and the modified Sombor index (mSO) are two closely related vertex-degree-based graph invariants. Both were introduced in the 2020s, and have already found a variety of chemical, physicochemical, and network-theoretical applications. In this paper, we examine the product SO. mSO and determine its main properties. It is found that the structure-dependence of this product is fully different from that of either SO or mSO. Lower and upper bounds for SO. mSO are established and the extremal graphs are characterized. For connected graphs, the minimum value of the product SO. mSO is the square of the number of edges. In the case of trees, the maximum value pertains to a special type of eclipsed sun graph, trees with a single branching point