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.