相关论文: The general Albertson irregularity index of graphs

The topic is the average order of a connected induced subgraph of a graph. 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 $n$, is…

The $\sigma$-irregularity index of a graph is defined as the sum of squared degree differences over all edges and provides a sensitive measure of structural heterogeneity. In this paper, we study the problem of maximizing $\sigma(T)$ among…

The eccentricity of a vertex is the maximum distance from it to another vertex and the average eccentricity $ecc (G)$ of a graph $G$ is the mean value of eccentricities of all vertices of $G$. The average eccentricity is deeply connected…

The Harary index of a graph is defined as the sum of reciprocals of distances between all pairs of vertices of the graph. In this paper we provide an upper bound of the Harary index in terms of the vertex or edge connectivity of a graph. We…

The distinguishing number (index) $D(G)$ ($D'(G)$) of a graph $G$ is the least integer $d$ such that $G$ has an vertex (edge) labeling with $d$ labels that is preserved only by the trivial automorphism. It is known that for every graph $G$…

For a connected graph $G$, let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal matrix of the degrees of the vertices in $G$. The $A_{\alpha}$-matrix of $G$ is defined as \begin{align*} A_\alpha (G) = \alpha D(G) +…

For a graph $G$, the generalized adjacency matrix $A_\alpha(G)$ is the convex combination of the diagonal matrix $D(G)$ and the adjacency matrix $A(G)$ and is defined as $A_\alpha(G)=\alpha D(G)+(1-\alpha) A(G)$ for $0\leq \alpha \leq 1$.…

Let $G$ be a connected graph. The revised edge 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 reverse Wiener index of a connected graph $G$ is a variation of the well-known Wiener index $W(G)$ defined as the sum of distances between all unordered pairs of vertices of $G$. It is defined as $\Lambda(G)=\frac{1}{2}n(n-1)d-W(G)$,…

Let $G = (V,E)$ be a graph and $k \ge 0$ an integer. A $k$-independent set $S \subseteq V$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. With $\alpha_k(G)$ we denote the maximum cardinality of…

The eccentric connectivity index of a graph $G$ is $\xi^c(G) = \sum_{v \in V(G)}\varepsilon(v)\deg(v)$, and the eccentric distance sum is $\xi^d(G) = \sum_{v \in V(G)}\varepsilon(v)D(v)$, where $\varepsilon(v)$ is the eccentricity of $v$,…

We investigate the \textit{group irregularity strength} ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge…

Let $D=(V,A)$ be a digraphs without isolated vertices. A vertex-degree based invariant $I(D)$ related to a real function $\varphi$ of $D$ is defined as a summation over all arcs, $I(D) = \frac{1}{2}\sum_{uv\in A}{\varphi(d_u^+,d_v^-)}$,…

For a connected graph $G$, an instance $I$ is a set of pairs of vertices and a corresponding routing $R$ is a set of paths specified for all vertex-pairs in $I$. Let $\mathfrak{R}_I$ be the collection of all routings with respect to $I$.…

We investigate the \textit{group irregularity strength} ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group $\gr$ of order $s$, there exists a function $f:E(G)\rightarrow \gr$ such that the sums of edge…

Let $G$ be a graph of order $n$, and let $A(G)$ and $D(G)$ be the adjacency matrix and the degree matrix of $G$ respectively. Define the convex linear combinations $A_\alpha (G)$ of $A (G)$ and $D (G) $ by $$A_\alpha (G)=\alpha…

We prove the following sharp estimate for the number of spanning trees of a graph in terms of its vertex-degrees: a simple graph $G$ on $n$ vertices has at most $(1/n^{2}) \prod_{v \in V(G)} (d(v)+1)$ spanning trees. This result is tight…

In this study we are interested mainly in investigating the relations between two graph irregularity measures which are widely used for structural irregularity characterization of connected graphs. Our study is focused on the comparison and…

Let $G=(V,E)$ be a connected graph, where $V=\{v_1, v_2, \cdots, v_n\}$ and $m=|E|$. $d_i$ will denote the degree of vertex $v_i$ of $G$, and $\Delta=\max_{1\leq i \leq n} d_i$. The ABC matrix of $G$ is defined as $M(G)=(m_{ij})_{n \times…

Let $G = (V, E)$ be a graph. The first Zagreb index of a graph $G$ is defined as $\sum_{u \in V} d^2(u)$, where $d(u)$ is the degree of vertex $u$ in $G$. Using the P\'{o}lya-Szeg\H{o} inequality, we in this paper present the first Zagreb…