Sum Index, Difference Index and Exclusive Sum Number of Graphs

Abstract We consider two recent conjectures made by Harrington, Henninger-Voss, Karhadkar, Robinson and Wong concerning relationships between the sum index, difference index exclusive number of graphs. One conjecture posits an exact relationship first invariants; we show that in fact predicted value may be arbitrarily far from truth either direction. In process establish some new bounds on both index. The other conjecture, can exceed large amount, follows known, but non-constructive, results; give explicit construction demonstrating it. Simultaneously with preprint this paper appearing, Harrington et al. updated their counterexamples to conjecture; however, only a discrepancy 1, one They therefore modified equality inequality; our results is still false general.