相关论文: The general Albertson irregularity index of graphs

Let $d_G(v)$ be the degree of the vertex $v$ in a graph $G$. The Sombor index of $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, which is a new degree-based topological index introduced by Gutman. Let…

The topic is the average order $A(G)$ of a connected induced subgraph of a graph $G$. This generalizes, to graphs in general, the average order of a subtree of a tree. In 1984, Jamison proved that the average order, over all trees of order…

The {\it total irregularity} of a simple undirected graph $G$ is defined as ${\rm irr}_t(G) =$ $\frac{1}{2}\sum_{u,v \in V(G)}$ $\left| d_G(u)-d_G(v) \right|$, where $d_G(u)$ denotes the degree of a vertex $u \in V(G)$. Obviously, ${\rm…

The total $\sigma$-irregularity is given by $ \sigma_t(G) = \sum_{\{u,v\} \subseteq V(G)} \left(d_G(u) - d_G(v)\right)^2, $ where $d_G(z)$ indicates the degree of a vertex $z$ within the graph $G$. It is known that the graphs maximizing…

Let $G$ be a connected graph. Let $N(G)$ and $S(G)$ be the number of connected sets of $G$ and the sum of the orders of these connected sets of $G$, respectively. Then $A(G)=\frac{S(G)}{N(G)}$ is called the average order of a connected set…

Recently, Gutman defined a new vertex-degree-based graph invariant, named the Sombor index $SO$ of a graph $G$, and is defined by $$SO(G)=\sum_{uv\in E(G)}\sqrt{d_G(u)^2+d_G(v)^2},$$ where $d_G(v)$ is the degree of the vertex $v$ of $G$. In…

A (molecular) graph in which all vertices have the same degree is known as a regular graph. According to Gutman, Hansen, and M\'elot [J. Chem. Inf. Model. 45 (2005) 222-230], it is of interest to measure the irregularity of nonregular…

Let $G=(V,E)$ be a simple connected graph with vertex set $V(G)$ and edge set $E(G)$. The third atom-bond connectivity index, $ABC_3$ index, of $G$ is defined as $ABC_3(G)=\sum\limits_{uv\in E(G)}\sqrt{\frac{e(u)+e(v)-2}{e(u)e(v)}}$, where…

An edge irregular total $k$-labelling $f : V(G)\cup E(G)\rightarrow \{1,2,\dots,k\}$ of a graph $G$ is a labelling of the vertices and the edges of $G$ in such a way that any two different edges have distinct weights. The weight of an edge…

In 2021, the Sombor index was introduced by Gutman, which is a new degree-based topological molecular descriptors. The Sombor index of a graph $G$ is defined as $SO(G) =\sum_{uv\in E(G)}\sqrt{d^2_G(u)+d^2_G(v)}$, where $d_G(v)$ is the…

In this paper, we investigate The relationship between the Albertson index and the first Zagreb index for trees. For a tree $T=(V,E)$ with $n=|V|$ vertices and $m=|E|$ edges, we provide several bounds and exact formulas for these two…

The general Sombor index of $G$ is defined as $SO_{\alpha}(G)= \sum_{uv\in G}\left(d^2_{G}(u)+d^2_{G}(v)\right)^{\alpha}$. For $0<\alpha<1$, we have the upper bound of $SO_{\alpha}(G)$ on unicyclic graphs with a fixed diameter, and the…

We establish sharp extremal bounds on the Albertson and Sigma irregularity indices for trees with prescribed degree sequences, with emphasis on caterpillar trees as key extremal configurations. New lower and upper bounds are derived in…

For a simple connected graph $G=(V,E)$, let $d(u)$ be the degree of the vertex $u$ of $G$. The general Sombor index of $G$ is defined as $$SO_{\alpha}(G)=\sum_{uv\in E} \left[d(u)^2+d(v)^2\right]^\alpha$$ where $SO(G)=SO_{0.5}(G)$ is the…

The average distance of a vertex $v$ of a connected graph $G$ is the arithmetic mean of the distances from $v$ to all other vertices of $G$. The proximity $\pi(G)$ and the remoteness $\rho(G)$ of $G$ are the minimum and the maximum of the…

Let $G$ be a connected finite graph with vertex set $V(G)$. The eccentricity $e(v)$ of a vertex $v$ is the distance from $v$ to a vertex farthest from $v$. The average eccentricity of $G$ is defined as $\frac{1}{|V(G)|}\sum_{v \in…

In this paper, we provide the irregularity properties of trees with strong support vertex by analyzing two prominent topological indices: the Albertson index and the Sigma index. We further establish extremal bounds for both indices across…

Let $G=(V,E)$ be a simple connected graph and $d_i$ be the degree of its $i$th vertex. In a recent paper [J. Math. Chem. 46 (2009) 1369-1376] the first geometric-arithmetic index of a graph $G$ was defined as $$GA_1=\sum_{ij\in E}\frac{2…

Let $G$ be a graph. Denote by $d_x$, $E(G)$, and $D(G)$ the degree of a vertex $x$ in $G$, the set of edges of $G$, and the degree set of $G$, respectively. This paper proposes to investigate (both from mathematical and applications points…

The harmonic index of a graph $G$ is defined as the sum of weights $\frac{2}{deg(v) + deg(u)}$ of all edges $uv$ of $E (G)$, where $deg (v)$ denotes the degree of a vertex $v$ in $V (G)$. In this note we generalize results of [L. Zhong, The…