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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. doi: 10.22108/toc.2014.5986
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. doi: 10.22108/toc.2014.5986
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. doi: 10.22108/toc.2014.5986
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. doi: 10.22108/toc.2014.5986

Complete solution to a conjecture of Zhang-Liu-Zhou

Article 21, Volume 3, Issue 4, December 2014, Page 55-58  XML PDF (473 K)
Document Type: Research Paper
DOI: 10.22108/toc.2014.5986
Authors
Mostafa Tavakoli 1; F. Rahbarnia1; M. Mirzavaziri1; A. R. Ashrafi2
1Ferdowsi University of Mashhad
2University of Kashan
Abstract
‎‎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.
Keywords
Eccentric connectivity index; tricyclic graph; tetracyclic graph; graph operation
Main Subjects
05A15 Exact enumeration problems, generating functions; 05A20 Combinatorial inequalities; 05C05 Trees; 05C12 Distance in graphs
References
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