20 research outputs found

    The quotients between the (revised) Szeged index and Wiener index of graphs

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    Let Sz(G),Sz∗(G)Sz(G),Sz^*(G) and W(G)W(G) be the Szeged index, revised Szeged index and Wiener index of a graph G.G. In this paper, the graphs with the fourth, fifth, sixth and seventh largest Wiener indices among all unicyclic graphs of order n⩾10n\geqslant 10 are characterized; as well the graphs with the first, second, third, and fourth largest Wiener indices among all bicyclic graphs are identified. Based on these results, further relation on the quotients between the (revised) Szeged index and the Wiener index are studied. Sharp lower bound on Sz(G)/W(G)Sz(G)/W(G) is determined for all connected graphs each of which contains at least one non-complete block. As well the connected graph with the second smallest value on Sz∗(G)/W(G)Sz^*(G)/W(G) is identified for GG containing at least one cycle.Comment: 25 pages, 5 figure

    Some graphs with extremal Szeged index

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    The extremal unicyclic graphs of the revised edge Szeged index with given diameter

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    Let GG be a connected graph. The revised edge Szeged index of GG is defined as Sze∗(G)=∑e=uv∈E(G)(mu(e∣G)+m0(e∣G)2)(mv(e∣G)+m0(e∣G)2)Sz^{\ast}_{e}(G)=\sum\limits_{e=uv\in E(G)}(m_{u}(e|G)+\frac{m_{0}(e|G)}{2})(m_{v}(e|G)+\frac{m_{0}(e|G)}{2}), where mu(e∣G)m_{u}(e|G) (resp., mv(e∣G)m_{v}(e|G)) is the number of edges whose distance to vertex uu (resp., vv) is smaller than the distance to vertex vv (resp., uu), and m0(e∣G)m_{0}(e|G) is the number of edges equidistant from both ends of ee, respectively. In this paper, the graphs with minimum revised edge Szeged index among all the unicyclic graphs with given diameter are characterized.Comment: arXiv admin note: text overlap with arXiv:1805.0657
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