Ordering graphs with large eccentricity-based topological indices

Abstract For a connected graph, the first Zagreb eccentricity index $\xi _{1}$ ? 1 is defined as sum of squares eccentricities all vertices, and second _{2}$ 2 products pairs adjacent vertices. In this paper, we mainly present different universal approach to determine upper bounds respectively on indices trees, unicyclic graphs bicyclic graphs, characterize these corresponding extremal which extend ordering results in (Du et al. Croat. Chem. Acta 85:359–362, 2012; Qi Discrete Appl. Math. 233:166–174, 2017; Li Zhang Comput. 352:180–187, 2019). Specifically, n -vertex trees with i -th largest for up $\lfloor n/2+1 \rfloor $ ? n / + ? compared three 2012), j n/2-1 ? 2n/5+1 5 two (Qi 2017), n/2-2\rfloor 2n/15+1\rfloor 15 (Li 2019), where $n\ge 6$ ? 6 . More importantly, propose kinds functions eccentricity-based topological indices, can yield more general simultaneously some classes indices. As applications, obtain about average eccentric connectivity graphs.