Seminár z teórie grafov - Katarína Hriňáková (22.11.2018)

vo štvrtok 22.11.2018 o 9:50 hod. v miestnosti M/213

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Prednášajúci: Katarína Hriňáková (STU Bratislava)

Názov: An inequality between variable Wiener index and variable Szeged index

Termín: 22.11.2018, 9:50 hod., M/213

Abstrakt:
For a simple graph $G=(V,E)$, the Wiener index is defined as the sum of distances between all pairs of vertices, i.e. $W(G)=\sum_{\{u, v\} \subseteq V(G)} d(u,v)$. The Szeged index is defined as $\Sz(G)= \sum_{e=ij \in E}n_{e}(i)\cdot n_{e}(j)$, where $n_{e}(i)$ is the number of vertices strictly closer to $i$ than $j$, and analogously, $n_{e}(j)$ is the number of vertices strictly closer to $j$. A well-know inequality between the Szeged and Wiener indices says that $\Sz(G)=\sum_{e=ij\in E(G)} n_e(i)n_e(j) \ge \sum_{\left\{u,v\right\}} d(u,v)= W(G)$ for every graph~$G$. In the past, variable variations of the standard topological indices were defined. Following this line, we study a natural generalisation of the above inequality, namely $\sum_{e=ij\in E(G)} (n_e(i)n_e(j))^\alpha \ge \sum_{\left\{u,v\right\}} d(u,v)^\alpha$. We show that for all trees the inequality is true if $\alpha>1$, and the opposite inequality holds if $0\le \alpha<1$. In fact, the first result also holds for bipartite graphs and for graphs on $n$ vertices with at most $n+3$ edges, but the opposite one does not. For general graphs we solve also the case $\alpha<0$ and we present interesting conjectures. Observe, that both the sums are interesting on their own, and in accordance with the usual terminology they can be called the variable Szeged and variable Wiener indices.

This is a joint work with Martin Knor and Riste Škrekovski.

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