Nonstandard Graphs, Revised

Tricyclic graphs with maximal revised Szeged index

The revised Szeged index of a graph $G$ is defined as $Sz^*(G)=\sum_{e=uv \in E}(n_u(e)+ n_0(e)/2)(n_v(e)+ n_0(e)/2),$ where $n_u(e)$ and $n_v(e)$ are, respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$, and $n_0(e)$ is the number of vertices equidistant to $u$ and $v$. In this paper, we give an upper bound of the revised Szeged index for a connected tricyclic graph, and also characterize those graphs that achieve the upper bound.

Steiner (revised) Szeged index of graphs

The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. For a connected graph $G$ of order at least 2 and $S\subseteq V(G)$, the Steiner distance $d_G(S)$ of the set $S$ of vertices in $G$ is the minimum size of a connected subgraph whose vertex set contains or connects $S$. In this paper, we introduce the concept of the Steiner (revised) Szeged index ($rSz_k(G)$) $Sz_k(G)$ of a graph $G$, which is a natural generalization of the well-known (revised) Szeged index of chemical use. We determine the $Sz_k(G)$…

图的（修正）Szeged指数与Wiener指数之间的商

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

本文刻画了单圈图和双圈图中Wiener指数极值图，并基于此研究了（修正）Szeged指数与Wiener指数之商的下界及极值图。

Bicyclic graphs with maximal revised Szeged index

The revised Szeged index $Sz^*(G)$ is defined as $Sz^*(G)=\sum_{e=uv \in E}(n_u(e)+ n_0(e)/2)(n_v(e)+ n_0(e)/2),$ where $n_u(e)$ and $n_v(e)$ are, respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number of vertices of $G$ lying closer to vertex $v$ than to vertex $u$, and $n_0(e)$ is the number of vertices equidistant to $u$ and $v$. Hansen used the AutoGraphiX and made the following conjecture about the revised Szeged index for a connected bicyclic graph $G$ of order $n \geq 6$: $$ Sz^*(G)\leq \{{array}{ll} (n^3+n^2-n-1)/4,& {if $n$ is odd}, (n…

A Turán Type Problem Concerning the Powers of the Degrees of a Graph (revised)

For a graph $G$ whose degree sequence is $d_{1},..., d_{n}$, and for a positive integer $p$, let $e_{p}(G)=\sum_{i=1}^{n}d_{i}^{p}$. For a fixed graph $H$, let $t_{p}(n,H)$ denote the maximum value of $e_{p}(G)$ taken over all graphs with $n$ vertices that do not contain $H$ as a subgraph. Clearly, $t_{1}(n,H)$ is twice the Turán number of $H$. In this paper we consider the case $p>1$. For some graphs $H$ we obtain exact results, for some others we can obtain asymptotically tight upper and lower bounds, and many interesting cases remain open.

给定直径的单圈图中修正边 Szeged 指标的极值图

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

本文在给定直径的单圈图类中，完全刻画了使修正边 Szeged 指标达到最小值的极值图结构。

The (revised) Szeged index and the Wiener index of a nonbipartite graph

Hansen et. al. used the computer programm AutoGraphiX to study the differences between the Szeged index $Sz(G)$ and the Wiener index $W(G)$, and between the revised Szeged index $Sz^*(G)$ and the Wiener index for a connected graph $G$. They conjectured that for a connected nonbipartite graph $G$ with $n \geq 5$ vertices and girth $g \geq 5,$ $ Sz(G)-W(G) \geq 2n-5. $ Moreover, the bound is best possible as shown by the graph composed of a cycle on 5 vertices, $C_5$, and a tree $T$ on $n-4$ vertices sharing a single vertex. They also conjectured that for a connected nonbipartite graph $G$ with …

A revised Moore bound for mixed graphs

The degree-diameter problem seeks to find the maximum possible order of a graph with a given (maximum) degree and diameter. It is known that graphs attaining the maximum possible value (the Moore bound) are extremely rare, but much activity is focused on finding new examples of graphs or families of graph with orders approaching the bound as closely as possible. There has been recent interest in this problem as it applies to mixed graphs, in which we allow some of the edges to be undirected and some directed. A 2008 paper of Nguyen and Miller derived an upper bound on the possible number of ve…

On extremal cacti with respect to the edge revised Szeged index

Let $G$ be a connected graph. The edge revised Szeged index of $G$ is defined as $Sz^{\ast}_{e}(G)=\sum\limits_{e=uv\in E(G)}(m_{u}(e|G)+\frac{m_{0}(e|G)}{2})(m_{v}(e|G)+\frac{m_{0}(e|G)}{2})$, where $m_{u}(e|G)$ (resp., $m_{v}(e|G)$) is the number of edges whose distance to vertex $u$ (resp., $v$) is smaller than the distance to vertex $v$ (resp., $u$), and $m_{0}(e|G)$ is the number of edges equidistant from both ends of $e$. In this paper, we give the minimal and the second minimal edge revised Szeged index of cacti with order $n$ and $k$ cycles, and all the graphs that achieve the minimal …

Proceedings of the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018)

These are the revised accepted papers from the 26th International Symposium on Graph Drawing and Network Visualization (GD 2018), Barcelona, Spain, September 26 - September 28, 2018. Proceedings are also to be published by Springer in the Lecture Notes in Computer Science series.

Linkedness of Cartesian products of complete graphs

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least $2k$ vertices is {\it $k$-linked} if, for every set of $2k$ distinct vertices organised in arbitrary $k$ pairs of vertices, there are $k$ vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product $K^{d_{1}+1}\times K^{d_{2}+1}$ of complete graphs $K^{d_{1}+1}$ and $K^{d_{2}+1}$ is $\floor{(d_{1}+d_{2})/2}$-linked for $d_{1},d_{2}\ge 2$, and this is best possible. %A polytope is said to be {\it $k$-linked} if its graph is $k$-linked. This result is connec…

The Matrix-Forest Theorem and Measuring Relations in Small Social Groups

We propose a family of graph structural indices related to the Matrix-forest theorem. The properties of the basic index that expresses the mutual connectivity of two vertices are studied in detail. The derivative indices that measure "dissociation," "solitariness," and "provinciality" of vertices are also considered. A nonstandard metric on the set of vertices is introduced, which is determined by their connectivity. The application of these indices in sociometry is discussed.

列表平方着色猜想对二部平面图及其线图不成立

The List Square Coloring Conjecture fails for bipartite planar graphs and their line graphs

本文构造二部平面图及其线图等反例，否定列表平方着色猜想的多个开放情形与平面版本，并改进有界最大度非可选图构造。