Extremal graphs of given parameters with respect to the eccentricity distance sum and the eccentric connectivity index

• Published in: Discrete Applied Mathematics Vol. 254; pp. 204 - 221
• Main Authors: Zhang, Huihui, Li, Shuchao, Xu, Baogen
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 15.02.2019; Elsevier BV
• Subjects: Cacti; Connectivity; Diameter; Eccentric connectivity index; Eccentricity; Eccentricity distance sum; Graphs; Independence number; Lower bounds; Minimum degree; Parameters; Upper bounds; Eccentricity distance sum; Minimum degree; Eccentric connectivity index; Diameter; Independence number
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2018.07.013

Given a connected graph G=(VG,EG), the eccentricity distance sum (EDS) of G is defined as ξd(G)=∑{u,v}⊆VG(ε(u)+ε(v))dG(u,v) and the eccentric connectivity index (ECI) of G is defined as ξc(G)=∑v∈VGε(v)dG(v), where ε(v),dG(v) and dG(u,v) are the eccentricity of v, the degree of v and the distance between u and v in G, respectively. In this paper, some extremal problems on the EDS and the ECI of graphs with given parameters are considered. Firstly, ordering the n-vertex graphs of diameter 2 with respect to the EDS (resp. the ECI) is established. Secondly, the sharp lower bound on the EDS of graphs with given minimum degree is determined. Moreover, the sharp upper bound on the ECI of graphs of diameter 2 with given minimum degree and the sharp lower bound on the ECI of cacti with given radius are established, respectively. Finally sharp upper and lower bounds on the difference between EDS and ECI among all n-vertex graphs of diameter 2 are determined. Thereafter the sharp lower bound on the difference between EDS and ECI among all n-vertex graphs of diameter 2 with given minimum degree (resp. connectivity, edge-connectivity and independence number) is determined.