On the Exponential Atom-Bond Connectivity Index of Graphs - Kinkar Chandra Das (Mathematics)
Several topological indices are possibly the most widely applied graph-based molecular structure descriptors in chemistry and pharmacology. The capacity of topological indices to discriminate is a crucial component of their study. In light of this, the literature has introduced the exponential vertex-degree-based topological index. The exponential atom-bond connectivity index is defined as follows: <semantics>eABC=eABC(Υ)=<mstyle displaystyle="true"><munder>∑vivj∈E(Υ)</munder></mstyle><mspace width="0.166667em"></mspace>e<msqrt><mstyle scriptlevel="0" displaystyle="true">di+dj−2di<mspace width="0.166667em"></mspace>dj</mstyle></msqrt>,</semantics> where <semantics>di</semantics> is the degree of the vertex <semantics>vi</semantics> in <semantics>Υ</semantics>. In this paper, we prove that the double star <semantics>DSn−3,1</semantics> is the second maximal graph with respect to the <semantics>eABC</semantics> index of trees of order n. We give an upper bound on <semantics>eABC</semantics> of unicyclic graphs of order n and characterize the maximal graphs. The graph <semantics>K1∨(P3∪(n−4)K1)</semantics> gives the maximal graph with respect to the <semantics>eABC</semantics> index of bicyclic graphs of order n. We present several relations between <semantics>eABC(Υ)</semantics> and <semantics>ABC(Υ)</semantics> of graph <semantics>Υ</semantics>. Finally, we provide a conclusion summarizing our findings and discuss potential directions for future research.