相关论文: Chebyshev's Sum Inequality and the Zagreb Indices …

For a simple graph $G$ with $n$ vertices and $m$ edges, the first Zagreb index and the second Zagreb index are defined as $M_1(G)=\sum_{v\in V}d(v)^2 $ and $M_2(G)=\sum_{uv\in E}d(u)d(v)$. In \cite{VGFAD}, it was shown that if a connected…

Let $G=(V,E)$ be a simple graph with $n = |V|$ vertices and $m = |E|$ edges. The first and second Zagreb indices are among the oldest and the most famous topological indices, defined as $M_1 = \sum_{i \in V} d_i^2$ and $M_2 = \sum_{(i, j)…

Let ${\mathcal G}_n$ be the set of class of graphs of order $n$. The first Zagreb index $M_1(G)$ is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index $M_2(G)$ is equal to the sum of the products of the…

The first Zagreb index $M_{1}(G)$ is equal to the sum of squares of the degrees of the vertices, and the second Zagreb index $M_{2}(G)$ is equal to the sum of the products of the degrees of pairs of adjacent vertices of the underlying…

Topological relations between three degree-based invariants of a connected graph G are investigated. We present novel inequalities including M1(G), M2(G) and F(G), and show that in all cases equality holds if G is a regular or a semiregular…

The second Zagreb index of a graph G is denoted by $M_2(G)=\sum_{uv\in E(G)}d(u)d(v)$. In this paper, we investigate properties of the extremal graphs with the maximum second Zagreb indices with given graphic sequences, in particular…

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…

The first Zagreb index of a graph $G$ is the sum of squares of the vertex degrees in a graph and the second Zagreb index of $G$ is the sum of products of degrees of adjacent vertices in $G$. The imbalance of an edge in $G$ is the numerical…

The first Zagreb index of a graph $G$ is the sum of the square of every vertex degree, while the second Zagreb index is the sum of the product of vertex degrees of each edge over all edges. In our work, we solve an open question about…

Let $G$ be a graph with order $n(G)$, size $m(G)$, first Zagreb index $M_1(G)$, and second Zagreb index $M_2(G)$. More than twenty years ago, it was conjectured that $\frac{M_1(G)}{n(G)} \leq \frac{M_2(G)}{m(G)}$. Later, Hansen and…

This paper presents new lower bounds for the first general Zagreb index $Z_{\alpha}(G)$ involving two, three, and four arbitrary degrees of vertices of a simple graph $G$. For the special cases $\alpha = 2$ and $\alpha = -2$, the results…

We examine the quantity \[S(G) = \sum_{uv\in E(G)} \min(\text{deg } u, \text{deg } v)\] over sets of graphs with a fixed number of edges. The main result shows the maximum possible value of $S(G)$ is achieved by three different classes of…

In this paper, we focus on comparing the first and second Zagreb-Fermat eccentricity indices of graphs. We show that $$\frac{\sum_{uv\in E\left( G \right)}{\varepsilon_3\left( u \right) \varepsilon_3\left( v \right)}}{m\left( G \right)}…

Let $D=(V,A)$ be a digraphs without isolated vertices. The first Zagreb index of a digraph $D$ defined as a summation over all arcs, $M_1(D)=\frac{1}{2}\sum\limits_{uv\in A}(d^{+}_{u}+d^{-}_v)$, where $d^{+}_u$(resp. $d^{-}_u$) denotes the…

The complementary second Zagreb index of a graph $G$ is defined as $cM_2(G)=\sum_{uv\in E(G)}|(d_u(G))^2-(d_v(G))^2|$, where $d_u(G)$ denotes the degree of a vertex $u$ in $G$ and $E(G)$ represents the edge set of $G$. Let $G^*$ be a graph…

The hyper Zagreb index is a kind of extensions of Zagreb index, used for predicting physicochemical properties of organic compounds. Given a graph $G= (V(G), E(G))$, the first hyper-Zagreb index is the sum of the square of edge degree over…

Let $G = (V, E)$ be a graph. The first Zagreb index and the forgotten topological index of a graph $G$ are defined respectively as $\sum_{u \in V} d^2(u)$ and $\sum_{u \in V} d^3(u)$, where $d(u)$ is the degree of vertex $u$ in $G$. If the…

Bidirected graphs generalize directed and undirected graphs in that edges are oriented locally at every node. The natural notion of the degree of a node that takes into account (local) orientations is that of net-degree. In this paper, we…

The first Zagreb index $M_{1}$ of a graph is defined as the sum of the square of every vertex degree, and the second Zagreb index $M_{2}$ of a graph is defined as the sum of the product of vertex degrees of each pair of adjacent vertices.…

The aim of this paper is to obtain new inequalities for a large family of topological indices restricted to unicyclic graphs and to characterize the set of extremal unicyclic graphs with respect to them. This family includes variable first…