Extremal Trees of the Eccentric Connectivity Index

Abstract

Chemical indices are introduced to correlate chemical compounds’ physical properties with their structures. Among recently introduced such indices, the eccentric connectivity index of a graph \(G\) is defined as \(\xi^C(G) = \sum_{v \in V(G)} deg(v) ec(v)\), where \(deg(v)\) is the degree of a vertex \(v\) and \( ec(v)\) is its eccentricity. The extremal values of \(\xi^C(G)\) have been studied among graphs with various given parameters. In this note, we study trees with extremal values of the eccentric connectivity index with a given degree sequence. The extremal structures are identified; however, they are not unique.