Title of article :
Complete solution to a conjecture of Zhang-Liu-Zhou
Author/Authors :
توكلي, م. ferdowsi university of mashhad - Department of Mathematics, Ferdowsi University of Mashhad, P O Box 1159, Mashhad 91775, مشهد, ايران , رهبريان, ف. ferdowsi university of mashhad - Department of Mathematics, Ferdowsi University of Mashhad, P O Box 1159, Mashhad 91775, مشهد, ايران , ميرزاوزيري, م. ferdowsi university of mashhad - Department of Mathematics, Ferdowsi University of Mashhad, P O Box 1159, Mashhad 91775, مشهد, ايران , اشرفي, علي رضا university of kashan - Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Kashan, P O Box 51167, Kashan, كاشان, ايران
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 ,
Journal title :
Transactions on Combinatorics
Journal title :
Transactions on Combinatorics
