Extremal results on stepwise transmission irregular graphs

Properties of stepwise irregular graphs

Stepwise irregular (SI) graphs were introduced by Ivan Gutman recently in 2018 and in these graphs the difference between the degrees of any two adjacent vertices is exactly one. In this work, we show the existence of connected bicyclic SI graphs of order 5 and 9 which has been claimed to be non-existent by Gutman. We also show the existence of SI graphs of different order and cyclomatic numbers when there exist a SI graph with a vertex of degree 1 or 2. We give a characterization on the order of the connected tricyclic SI graphs. At the end, we investigate some properties of the SI graphs und…

关于 $k$-步不规则图的极值结果

Extremal results on $k$-stepwise irregular graphs

本文研究了 $k$-步不规则图的极值性质，证明了其存在性并界定了最大度数和图大小，引入了度复杂度的概念。

广义逐步传输不规则图

Generalized stepwise transmission irregular graphs

本研究引入广义逐步传输不规则（STI）图的概念，证明其二分性、2-边连通性等性质，确定树中仅星型图为广义STI图，给出直径界并构造多类相关图。

极值度不规则性界限研究：关于Albertson指数和Sigma指数在树和双图中的应用

Extremal Degree Irregularity Bounds for Albertson and Sigma Indices in Trees and Bipartite Graphs

本文研究了树和双图中Albertson和Sigma指数的极值界限，推导了不规则性度量的显式公式和上界，通过度数序列的不等式推进了对双图极值不规则性行为的理解。

The general Albertson irregularity index of graphs

We introduce the general Albertson irregularity index of a connected graph $G$ and define it as $A_{p}(G) =(\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}}$, where $p$ is a positive real number and $d(v)$ is the degree of the vertex $v$ in $G$. The new index is not only generalization of the well-known Albertson irregularity index and $σ$-index, but also it is the Minkowski norm of the degree of vertex. We present lower and upper bounds on the general Albertson irregularity index. In addition, we study the extremal value on the general Albertson irregularity index for trees of given order. Final…

关于具有最大维纳指数的传输不规则树

On the transmission irregular trees with the maximum Wiener index

本文研究了在给定阶数的传输不规则树中最大化维纳指数的问题，针对奇数和几乎所有偶数阶数给出了唯一解。