Tavakoli, M., Rahbarnia, F., Mirzavaziri, M., Ashrafi, A. (2014). Complete solution to a conjecture of Zhang-Liu-Zhou. Transactions on Combinatorics, 3(4), 55-58.
Mostafa Tavakoli; F. Rahbarnia; M. Mirzavaziri; A. R. Ashrafi. "Complete solution to a conjecture of Zhang-Liu-Zhou". Transactions on Combinatorics, 3, 4, 2014, 55-58.
Tavakoli, M., Rahbarnia, F., Mirzavaziri, M., Ashrafi, A. (2014). 'Complete solution to a conjecture of Zhang-Liu-Zhou', Transactions on Combinatorics, 3(4), pp. 55-58.
Tavakoli, M., Rahbarnia, F., Mirzavaziri, M., Ashrafi, A. Complete solution to a conjecture of Zhang-Liu-Zhou. Transactions on Combinatorics, 2014; 3(4): 55-58.
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.
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