complete solution to a conjecture of zhang-liu-zhou

‎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.