On the eccentric connectivity index of trees with given domination number

Let G be a simple connected finite graph. The eccentric connectivity index (ECI) of G isdefined as xi(c)(G) = Sigma(v is an element of V(G)) epsilon(G)(V)d(G)(v), where epsilon G(v) is the eccentricity of v, dG(v) is the degree of v. We denote the set of trees with order n and domination number gamma by T-n,T-gamma . In this paper, the extremal trees having the minimal ECI among Tn,gamma are determined. The tree among Tn,gamma satisfying 2 <= gamma <= inverted left perpendicular n/3 inverted right perpendicular having the maximal ECI is also characterized. For inverted left perpendicular n/3 inverted right perpendicular <= gamma <= 2n, the tree among all caterpillars with domination number gamma having the maximal ECI is determined. (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.