Total eccentricity index of graphs with fixed number of pendant or cut vertices

Wiener index of graphs with fixed number of pendant or cut vertices

The Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut vertices.

Cut and pendant vertices and the number of connected induced subgraphs of a graph

A vertex whose removal in a graph $G$ increases the number of components of $G$ is called a cut vertex. For all $n,c$, we determine the maximum number of connected induced subgraphs in a connected graph with order $n$ and $c$ cut vertices, and also characterise those graphs attaining the bound. Moreover, we show that the cycle has the smallest number of connected induced subgraphs among all cut vertex-free connected graphs. The general case $c>0$ remains an open task. We also characterise the extremal graph structures given both order and number of pendant vertices, and establish the correspon…

On the extremal total reciprocal edge-eccentricity of trees

The total reciprocal edge-eccentricity is a novel graph invariant with vast potential in structure activity/property relationships. This graph invariant displays high discriminating power with respect to both biological activity and physical properties. If $G=(V_G,E_G)$ is a simple connected graph, then the total reciprocal edge-eccentricity (REE) of $G$ is defined as $ξ^{ee}(G)=\sum_{uv\in E_G}(1/\varepsilon_G(u)+1/\varepsilon_G(v))$, where $\varepsilon_G(v)$ is the eccentricity of the vertex $v$. In this paper we first introduced four edge-grafting transformations to study the mathematical p…

On local antimagic chromatic number of graphs with cut-vertices

An edge labeling of a connected graph $G = (V, E)$ is said to be local antimagic if it is a bijection $f:E \to\{1,\ldots ,|E|\}$ such that for any pair of adjacent vertices $x$ and $y$, $f^+(x)\not= f^+(y)$, where the induced vertex label $f^+(x)= \sum f(e)$, with $e$ ranging over all the edges incident to $x$. The local antimagic chromatic number of $G$, denoted by $χ_{la}(G)$, is the minimum number of distinct induced vertex labels over all local antimagic labelings of $G$. In this paper, the sharp lower bound of the local antimagic chromatic number of a graph with cut-vertices given by pend…

Cut Tree Structures with Applications on Contraction-Based Sparsification

We introduce three new cut tree structures of graphs $G$ in which the vertex set of the tree is a partition of $V(G)$ and contractions of tree vertices satisfy sparsification requirements that preserve various types of cuts. Recently, Kawarabayashi and Thorup \cite{Kawarabayashi2015a} presented the first deterministic near-linear edge-connectivity recognition algorithm. A crucial step in this algorithm uses the existence of vertex subsets of a simple graph $G$ whose contractions leave a graph with $\tilde{O}(n/δ)$ vertices and $\tilde{O}(n)$ edges ($n := |V(G)|$) such that all non-trivial min-…

反转带长悬挂路径的Mostar线图不等式

Reversing the Mostar line-graph inequality with long pendant paths

本文研究悬挂路径长度对线图与原图Mostar指数差的影响，证明其仿射性，并解决Alex-Indulal问题，给出正圈数下最大度为3的无穷多反例及相等性条件。