University of Isfahan Transactions on Combinatorics 2251-8657 3 4 2014 12 01 Complete solution to a conjecture of Zhang-Liu-Zhou 55 58 5986 10.22108/toc.2014.5986 EN Mostafa Tavakoli Ferdowsi University of Mashhad F. Rahbarnia Ferdowsi University of Mashhad M. Mirzavaziri Ferdowsi University of Mashhad A. R. Ashrafi University of Kashan Journal Article 2014 01 20 ‎‎Let $d_{n,m}=\big[\frac{2n+1-\sqrt{17+8(m-n)}}{2}\big]$ and‎ ‎$E_{n,m}$ be the graph obtained from a path‎ ‎$P_{d_{n,m}+1}=v_0v_1 \cdots v_{d_{n,m}}$ by joining each vertex of‎ ‎$K_{n-d_{n,m}-1}$ to $v_{d_{n,m}}$ and $v_{d_{n,m}-1}$‎, ‎and by‎ ‎joining $m-n+1-{n-d_{n,m}\choose 2}$ vertices of $K_{n-d_{n,m}-1}$‎ ‎to $v_{d_{n,m}-2}$‎. ‎Zhang‎, ‎Liu and Zhou [On the maximal eccentric‎ ‎connectivity indices of graphs‎, ‎Appl‎. ‎Math‎. ‎J‎. ‎Chinese Univ.‎, ‎in‎ ‎press] conjectured that if $d_{n,m}\geqslant 3$‎, ‎then $E_{n,m}$‎ ‎is the graph with maximal eccentric connectivity index among all‎ ‎connected graph with $n$ vertices and $m$ edges‎. ‎In this note‎, ‎we‎ ‎prove this conjecture‎. ‎Moreover‎, ‎we present the graph with‎ ‎maximal eccentric connectivity index among the connected graphs‎ ‎with $n$ vertices‎. ‎Finally‎, ‎the minimum of this graph invariant‎ ‎in the classes of tricyclic and tetracyclic graphs are computed‎. https://toc.ui.ac.ir/article_5986_740f215bc6659e95ccaa77c44e50e504.pdf