Álvaro Martínez-Pérez, osé M. Rodríguez
The aim of this paper is to obtain new inequalities for a large family of generalizations of the Wiener Index and to characterize the set of extremal graphs with respect to them. Our main results provide upper and lower bounds for these topological indices on unicyclic graphs.
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In this article, we determine the maximum Wiener indices of unicyclic graphs with given number of vertices and matching number. We also characterize the extremal graphs. This solves an open problem of Du and Zhou.
The Wiener Index of Unicyclic Graphs with Girth and the Matching Number
本文研究了单圈图的Wiener指数随图的操作而变化的规律。这些结果被用来提出一个尖锐的下界的Wiener指数的顺序$n$周长和匹配数$β\ge \frac{3g}{2}$的单圈图。此外，我们还刻画了所有达到下界的极图。
On minimizing the Wiener index of unicyclic graphs with fixed girth and given degree sequence
针对固定围长与给定度序列的单圈图，研究依据质心位置提出三类 Wiener 指数最小化候选结构。
The \emph{Wiener index} is a widely studied topological index of graphs. One of the main problems in the area is to determine which graphs of given properties attain the extremal values of Wiener index. In this paper we resolve an open problem posed by Du in 2012. To this end we determine the unicyclic bipartite graphs with given size of parts having the maximum Wiener index. This completes the previous research in which the minimum case was solved.
Let $G=(V(G),E(G))$ be a molecular graph, where $V(G)$ and $E(G)$ are the sets of vertices (atoms) and edges (bonds). A topological index of a molecular graph is a numerical quantity which helps to predict the chemical/physical properties of the molecules. The Wiener, Wiener polarity and the terminal Wiener indices are the distance based topological indices. In this paper, we described a linear time algorithm {\bf(LTA)} that computes the Wiener index for acyclic graphs and extended this algorithm for unicyclic graphs. The same algorithms are modified to compute the terminal Wiener index and th…
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 Zagreb, variable sum exdeg, multiplicative second Zagreb and Narumi-Katayama indices. Our main results provide upper and lower bounds for these topological indices on unicyclic graphs, fixing or not the maximum degree or the number of pendant vertices.
Sum-connectivity indices of trees and unicyclic graphs of fixed maximum degree
在给定顶点数和最大度的条件下，分别得到了树集和单圈图集中图的最大和连通度指数，并确定了相应的极图.此外，我们还推导出了$n\ge 4$的具有前两个最大和连通度的n-顶点单圈图。
The largest $n-1$ Hosoya indices of unicyclic graphs
图$G$的Hosoya指标$Z (G)$定义为图$G$的边独立集的总数.本文推广了[J. Ou，On extremal unicyclic molecular graphs with maximum Hosoya index，\textit{Discrete Appl. Math.} 157（2009）391--397.]和[Y。是的，X。潘，H. Liu，Ordering unicyclic graphs with respect to Hosoya indices and Merrifield-Simmons indices，\textit{MATCH Commun. Math. Comput. Chem.} 59（2008）191--202.]并根据Hosoya指标对最大的$n-1$单圈图进行排序。
In the study of topological indices two negative correlations are well known: that between the number of subtrees and the Wiener index (sum of distances), and that between the Merrifield-Simmons index (number of independent vertex subsets) and the Hosoya index (number of independent edge subsets). That is, among a certain class of graphs, the extremal graphs that maximize one index usually minimize the other, and vice versa. In this paper, we first study the numbers of subtrees in unicyclic graphs and unicyclic graphs with a given girth, further confirming its opposite behavior to the Wiener i…
The Kirchhoff indices and the matching numbers of unicyclic graphs
一个连通图的Kirchhoff指标是图中所有无序顶点对之间的电阻距离之和。它在许多领域都有相当多的应用。本文确定了具有固定顶点数和匹配数的单圈图的最小Kirchhoff指标，并刻画了极图的特征。
Cut vertices are often used as a measure of nodes' importance within a network. They are those nodes whose failure disconnects a graph. Let N(G) be the number of connected induced subgraphs of a graph $G$. In this work, we investigate the maximum of N(G) where $G$ is a unicyclic graph with $n$ nodes of which $c$ are cut vertices. For all valid $n,c$, we give a full description of those maximal (that maximise N(.)) unicyclic graphs. It is found that there are generally two maximal unicyclic graphs. For infinitely many values of $n,c$, however, there is a unique maximal unicyclic graph with $n$ …
Alternative Proofs on the Indices of Cacti and Unicyclic Graphs with $n$ Vertices
令$H_n$是从星形$K_{1,n-1}$通过在悬垂顶点对之间添加$\lfloor \frac{n-1}{2}\rfloor$独立边而得到的仙人掌。设$K_{1,n-1}^+$是由星形图$K_{1,n-1}$通过附加一条边而得到的单圈图。本文给出了以下结果的另一种证明：在所有具有$n$顶点的仙人掌中，$H_n$是唯一的谱半径最大的仙人掌;在所有具有$n$顶点的单圈图中，$K_{1,n-1}^+$是唯一的谱半径最大的单圈图。证明了在所有顶点为$n$的奇圈图中，$H_n$是唯一一个谱半径最大的奇圈图.
On the Zagreb indices of the line graph and line cut-vertex graph of subdivision of unicyclic graphs
本文借助细分概念，研究环-星图这类单圈图的线图及线割点图的Zagreb指数与余指数。
Applications in chemistry motivated mathematicians to define different topological indices for different types of graphs. The Hyper-Zagreb index (HM) is an important tool as it integrates the first and the second Zagreb indices. In this paper, we characterize the trees and unicyclic graphs with the first four and first eight greatest HM-value, respectively.
Sombor index is a novel topological index, which was introduced by Gutman and defined for a graph $G$ as $SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}$, where $d_{u}=d_{G}(u)$ denotes the degree of vertex $u$ in graph $G$. Extremal problems on the Sombor index for trees with a given diameter has been considered by Chen et al. [H. Chen, W. Li, J. Wang, Extremal values on the Sombor index of trees, MATCH Commun. Math. Comput. Chem. 87 (2022) 23--49] and Li et al. [S. Li, Z. Wang, M. Zhang, On the extremal Sombor index of trees with a given diameter, Appl. Math. Comput. 416 (2022) 126…
F-index of a graph is the sum of the cube of the degrees of the vertices. In this paper, we investigate the F-indices of unicyclic graphs by introducing some transformation, and characterize the unicyclic graphs with the first five largest F-indices and the unicyclic graphs with the first two smallest F-indices, respectively.
Over all graphs (or unicyclic graphs) of a given order, we characterise those graphs that minimise or maximise the number of connected induced subgraphs. For each of these classes, we find that the graphs that minimise the number of connected induced subgraphs coincide with those that are known to maximise the Wiener index (the sum of the distances between all unordered pairs of vertices), and vice versa. For every $k$, we also determine the connected graphs that are extremal with respect to the number of $k$-vertex connected induced subgraphs. We show that, in contrast to the minimum which is…
The Wiener and Terminal Wiener indices of trees
Heydari \cite{heydari2013}给出了广义Bethe树的Wiener指标和终端Wiener指标的很好的公式。遗憾的是，这些公式存在一些误差。在本文中，我们纠正了这些错误，并在所有树的顺序$n$和最大程度$Δ$中，所有的树的最小终端Wiener指标的特点。
The Wiener index of a connected graph is the sum of the distance of all pairs of distinct vertices. It was introduced by Wiener in 1947 to analyze some aspects of branching by fitting experimental data for several properties of alkane compounds. Denote by $\mathcal{U}_{n,r}$ the set of unicyclic graphs with $n$ vertices and $r$ vertices of even degree. In this paper we present a structural result on the graphs in $\mathcal{U}_{n,r}$ with minimum Wiener index and completely characterize such graphs when $ r\leq \frac{n+3}{2}$.
Wiener Indices of Spiro and Polyphenyl Hexagonal Chains
连通图$G$的Wiener指数W（G）是G$中所有顶点对之间的距离之和。本文首先给出了一类无支链多螺环分子和多环芳烃的图--螺环和多苯基六方链的Wiener指数的递推或显式计算公式，然后建立了螺环六方链和多苯基六方链的Wiener指数之间的关系，确定了具有n个六边形的螺环六角链和聚苯六角链的Wiener指数极值，并刻画了它们的Wiener指数极值图.一个有趣的结果表明，所有六角链的Wiener指数的平均值正好是三个特殊六角链的Wiener指数的平均值，并且是元链的Wiener指数.
Contraction Formulas For Kirchhoff And Wiener Indices
我们建立了几个基尔霍夫指数的压缩公式。我们将Kirchhoff指标与其他一些度量化图不变量联系起来。当图是树时，通过连续应用我们的收缩公式，我们得到了新的Wiener指标公式，并得到了一些已知的Wiener指标公式，并给出了新的证明。
Ordering trees having small reverse Wiener indices
连通图$G$的反向Wiener指标是著名的Wiener指标$W(G)$的一个变体，Wiener指标定义为$G$的所有无序顶点对之间的距离之和。它被定义为$Λ(G)=\frac{1}{2}n(n-1)d-W(G)$，其中$n$是顶点数，$d$是$G$的直径。现在，我们确定第二和第三个最小的反向维纳指数的$n$ -顶点树和表征树的反向维纳指数达到这些值为$n\ge 6$（它已被称为明星是唯一的树与最小的反向维纳指数）。
The eccentric sequence of a connected graph $G$ is the nondecreasing sequence of the eccentricities of its vertices. The Wiener index of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. The unique trees that minimise the Wiener index among all trees with a given eccentric sequence were recently determined by the present authors. In this paper we show that these results hold not only for the Wiener index, but for a large class of distance-based topological indices which we term Wiener-type indices. Particular cases of this class include the hyper-Wiener index, the…
Wiener Indices of Minuscule Lattices
本文研究微小偏序集中序理想的分配格的维纳指数，给出了精确表达式，并分析了无限族中随机序理想间距离的渐近分布。
Topological indices are a class of numerical invariants that predict certain physical and chemical properties of molecules. Recently, two novel topological indices, named as Sombor index and reduced Sombor index, were introduced by Gutman, defined as $$SO(G)=\sum_{uv\in E(G)}\sqrt{d_{G}^{2}(u)+d_{G}^{2}(v)},$$ $$SO_{red}(G)=\sum_{uv\in E(G)}\sqrt{(d_{G}(u)-1)^{2}+(d_{G}(v)-1)^{2}},$$ where $d_{G}(u)$ denotes the degree of vertex $u$ in $G$. In this paper, our aim is to order the chemical trees, chemical unicyclic graphs, chemical bicyclic graphs and chemical tricyclic graphs with respect to So…
In this paper, we investigate the edge-Wiener index, the Szeged index, the edge-Szeged index, and the PI index, which are some of the most studied distance-based topological indices. As the main result we show that for benzenoid systems these indices can be computed in sub-linear time with respect to the number of vertices. More precisely, they can be computed in the time dependent on the length of the boundary cycle of a benzenoid system.
For a graph $G,$ we denote the number of connected subgraphs of $G$ by $F(G)$. For a tree $T$, $F(T)$ has been studied extensively and it has been observed that $F(T)$ has a reverse correlation with Wiener index of $T$. Based on that, we call $F(G),$ the core index of $G$. In this paper, we characterize the graphs which extremize the core index among all graphs on $n$ vertices with $k\geq 0$ connected components. We extend our study of core index to unicyclic graphs and connected graphs with fixed number of pendant vertices. We obtained the unicyclic graphs which extremize the core index over …
The Wiener index $W(G)$ is the sum of distances of all pairs of vertices of the graph $G$. The Wiener polarity index $W_{p}(G)$ of a graph $G$ is the number of unordered pairs of vertices $u$ and $v$ of $G$ such that the distance $d_{G}(u,v)$ between $u$ and $v$ is $3$. In this paper the Wiener and the Wiener polarity indices of sun graphs are computed. A relationship between those indices with some other topological indices are presented. Finally, we find the Hosoya (Wiener) polynomial for sun graphs.
A graph is maximal $k$-degenerate if each induced subgraph has a vertex of degree at most $k$ and adding any new edge to the graph violates this condition. In this paper, we provide sharp lower and upper bounds on Wiener indices of maximal $k$-degenerate graphs of order $n \ge k \ge 1$. A graph is chordal if every induced cycle in the graph is a triangle and chordal maximal $k$-degenerate graphs of order $n \ge k$ are $k$-trees. For $k$-trees of order $n \ge 2k+2$, we characterize all extremal graphs for the upper bound.
On the Wiener and Harary Indices of Generalized Splitting and Shadow-Splitting Graphs
研究连通图\(G\)的广义分裂图\(S_{p,q}(G)\)和影子分裂图\(H_{c,k}(G)\)，确定其维纳指数与哈拉里指数。
Distance Sequences to bound the Harary Index and other Wiener-type Indices of a Graph
本文利用距离序列方法，为包含 Wiener 指数、Harary 指数和超 Wiener 指数在内的广泛距离型拓扑指数建立一般界，并给出多类图上的锐利上下界。
Daisy cubes are a class of isometric subgraphs of the hypercubes Q n. Daisy cubes include some previously well known families of graphs like Fibonacci cubes and Lucas cubes. Moreover they appear in chemical graph theory. Two distance invariants, Wiener and Mostar indices, have been introduced in the context of the mathematical chemistry. The Wiener index W (G) is the sum of distance between all unordered pairs of vertices of a graph G. The Mostar index Mo(G) is a measure of how far G is from being distance balanced. In this paper we establish that the Wiener and the Mostar indices of a daisy c…
Corrigendum on Wiener index, Zagreb Indices and Harary index of Eulerian graphs
本文勘误指出欧拉图Wiener指数最大化原证明存在缺陷，实为Plesník结果的推论，并修正了两篇引用论文中的错误证明，给出了正确证明。
The first and the second Zagreb eccentricity index of a graph $G$ are defined as $E_1(G)=\sum_{v\in V(G)}\varepsilon_{G}(v)^{2}$ and $E_2(G)=\sum_{uv\in E(G)}\varepsilon_{G}(u)\varepsilon_{G}(v)$, respectively, where $\varepsilon_G(v)$ is the eccentricity of a vertex $v$. In this paper the invariants $E_1$, $E_2$, and the Wiener index are compared on graphs with diameter $2$, on trees, on a newly introduced class of universally diametrical graphs, and on Cartesian product graphs. In particular, if the diameter of a tree $T$ is not too big, then $W(T) \ge E_2(T)$ holds, and if the diameter of $…
Topological indices are parameters associated with graphs that have many applications in different areas such as mathematical chemistry. Among various topological indices, the Wiener index is classical \cite{w}. In this paper, we prove a formula for the Wiener index and more general $m$-Steiner Wiener index of an arbitrary graph $G$ in terms of the cardinalities of its twin classes. In particular, we will show that calculating these parameters for the graph $G$ can be reduced to calculating the same for a much smaller graph (in general) called the reduced graph of $G$. As applications of our m…
Five results on maximizing topological indices in graphs
本文证明了关于图形指数的五个结果，包括确定极值图、推广已有工作、提供替代证明，并解决两个猜想，涉及维纳指数、外部维纳指数、加权塞格德指数等。研究问题聚焦于给定匹配数或独立数等条件下的指数最大化，核心方法是推导与证明，主要贡献在于推广和解决相关猜想。
Besides the well known Wiener index, which sums up the distances between all the pairs of vertices, and the hyper-Wiener index, which includes also the squares of distances, the edge versions of both indices attracted a lot of attention in the recent years. In this paper we consider the edge-Wiener index and the edge-hyper-Wiener index of phenylenes, which represent an important class of molecular graphs. For an arbitrary phenylene, four quotient trees based on the elementary cuts are defined in a similar way as it was previously done for benzenoid systems. The computation of the edge-Wiener i…
The quotients between the (revised) Szeged index and Wiener index of graphs
本文刻画了单圈图和双圈图中Wiener指数极值图，并基于此研究了（修正）Szeged指数与Wiener指数之商的下界及极值图。
Recently, D. Vuki$\check{c}$evi$\acute{c}$ and J. Sedlar in \cite{Vuki} introduced an order "$\preceq$" on $\mathcal{T}_n$, the set of trees on $n$ vertices, such that the topological index $F$ of a graph is a function defined on the order set $\langle\mathcal{T}_n,\preceq\rangle$. It provides a new approach to determine the extremal graphs with respect to topological index $F$. By using the method they determined the common maximum and/or minimum graphs of $\mathcal{T}_n$ with respect to topological indices of Wiener type and anti-Wiener type. Motivated by their researches we further study th…
The Graovac-Pisanski index, also called the modified Wiener index, was introduced in 1991 and represents an extension of the original Wiener index, because it considers beside the distances in a graph also its symmetries. Similarly as Wiener in 1947 showed the correlation of the Wiener indices of the alkane series with the boiling points, in 2018 the connection between the Graovac-Pisanski index and the melting points of some hydrocarbon molecules was established. In this paper, we prove that the Graovac-Pisanski index of any connected bipartite graph as well as of any connected graph on an ev…
On the Wiener-like root-indices of graphs
本文研究图多项式根的图度量性质，证明修正多项式的解析结果，并引入新度量以比较拓扑指标，发现根度量在树上有高区分能力。
Let $I(G)$ be a topological index of a graph. If $I(G+e)<I(G)$ (or $I(G+e)>I(G)$, respectively) for each edge $e\not\in G$, then $I(G)$ is monotonically decreasing (or increasing, respectively) with the addition of edges. In this article, we present lower or upper bounds for some monotonic topological indices, including the Wiener index, the hyper-Wiener index, the Harary index, the connective eccentricity index, the eccentricity distance sum of bipartite graphs in terms of the number of cut edges, and characterize the corresponding extremal graphs, respectively.
Maximum Inverse Sum Indeg Index of Trees and Unicyclic Graphs with Fixed Diameter
本文研究了具有固定直径的树和单环图的最大逆向和入度指数，通过图变换方法确定了极值图结构。
In this paper we study the Szeged index of partial cubes and hence generalize the result proved by V. Chepoi and S. Klavžar, who calculated this index for benzenoid systems. It is proved that the problem of calculating the Szeged index of a partial cube can be reduced to the problem of calculating the Szeged indices of weighted quotient graphs with respect to a partition coarser than $Θ$-partition. Similar result for the Wiener index was recently proved by S. Klavžar and M. J. Nadjafi-Arani. Furthermore, we show that such quotient graphs of partial cubes are again partial cubes. Since the resu…
Wiener index and Harary index are two classic and well-known topological indices for the characterization of molecular graphs. Recently, Yu et al. \cite{YYSX} established some sufficient conditions for a graph to be pancyclic in terms of the edge number, the spectral radius and the signless Laplacian spectral radius of the graph. In this paper, we give some sufficient conditions for a graph being pancyclic in terms of the Wiener index, the Harary index, the distance spectral radius and the Harary spectral radius of a graph.
The comparison of two Zagreb-Fermat eccentricity indices
本文比较图的第一和第二 Zagreb-Fermat 离心率指数，证明无环图和单环图满足特定不等式，且该不等式不适用于至少含两个环的图。
Calculating the correlation coefficients of graph-theoretical indices
使用生成函数的方法，四个不同的图论指标，即独立顶点子集的数量，匹配的数量，子树的数量和维纳指数的相关系数，渐近确定随机根有序树。
Topological indices have important role in theoretical chemistry for QSPR researches. Among the all topological indices the Randić and the Zagreb indices have been used more considerably than any other topological indices in chemical and mathematical literature. Most of the topological indices as in the Randić and the Zagreb indices are based on the degrees of the vertices of a connected graph. Recently novel two degree concepts have been defined in graph theory; ev-degrees and ve-degrees. In this study we define ev-degree Zagreb index, ve-degree Zagreb indices and ve-degree Randić index by us…
The Wiener index is defined as the sum of distances between all unordered pairs of vertices in a graph. It is one of the most recognized and well-researched topological indices, which is on the other hand still a very active area of research. This work presents a natural continuation of the paper Mathematical aspects of Wiener index (Ars Math. Contemp., 2016) in which several interesting open questions on the topic were outlined. Here we collect answers gathered so far, give further insights on the topic of extremal values of Wiener index in different settings, and present further intriguing p…
The hyper Wiener index of one pentagonal carbon nanocone
本文的目的是计算单五边形碳纳米锥的拓扑指数之一。单五边形碳纳米锥是由一个五边形为核心，周围是多层六边形组成的，如果有n层六边形，则这种分子的图形记为G_n .本文的目的是明确地计算G_n的超Wiener指数.
In the paper we develop new methods for calculating the two well-known topological indices, the degree-distance and the Gutman index. Firstly, we prove that the Wiener index of a double vertex-weighted graph can be computed from the Wiener indices of weighted quotient graphs with respect to a partition of the edge set that is coarser than $Θ^*$-partition. This result immediately gives a method for computing the degree-distance of any graph. Next, we express the degree-distance and the Gutman index of an arbitrary phenylene by using its hexagonal squeeze and inner dual. In addition, it is shown…
Generalizations of Wiener polarity index and terminal Wiener index
在理论化学中，基于距离的分子结构描述符用于模拟化合物的物理，药理学，生物学和其他性质。我们引入广义Wiener极性指数$W_k (G)$作为$G$的无序顶点对${u, v}$的数量，使得$u$和$v$之间的最短距离$d (u, v)$是$k$（这实际上是Wiener多项式中的第$k$个系数）。对于$k = 3$，我们得到标准的Wiener极性指数。此外，我们推广的终端Wiener指数$TW_k (G)$之间的距离和所有对度$k$的顶点。对于$k = 1$，我们得到标准的终端Wiener指数。在本文中，我们描述了一个线性时间算法计算这些指标的树和部分立方体，并刻画极值树最大化广义Wiener极性指数和广义终端Wiener指数之间的所有给定顺序$n$的树。
On the Augmented Zagreb Index
拓扑指数在数学化学特别是定量结构-性质关系（QSPR）和定量结构-活性关系（QSAR）中起着重要的作用。最近的研究表明，增广萨格勒布指数（AZI）具有最好的相关能力，在几个拓扑指数。目前的研究的主要目的是建立一些数学性质的指数，或更准确地说，报告紧界的化学双环和化学单圈图的AZI。还得到了（补图连通的连通图的）AZI的一个Nordhaus-Gaddum型结果.
Simple Alcohols with the Lowest Normal Boiling Point Using Topological Indices
我们找到了具有给定碳原子数和最小正常沸点的简单饱和醇。预测的沸点的加权和的广义第一萨格勒布指数，第二萨格勒布指数，维纳指数顶点加权图，和一个简单的指数照顾的程度的碳原子入射到羟基。为了找到极端的醇分子，我们表征化学树的顺序$n$，最小化的第二萨格勒布指数和广义的第一萨格勒布指数的总和，也建立化学树，最小化的维纳指数在所有的化学树给定的顶点权重。
Let G be a simple connected graph having vertex set V and edge set E. The vertex-set and edge-set of G denoted by V(G) and E(G), respectively. The length of the smallest path between two vertices is called the distance. Mathematical chemistry is the area of research engaged in new application of mathematics in chemistry. In mathematics chemistry, we have many topological indices for any molecular graph, that they are invariant on the graph automorphism. In this research paper, we computing the Wiener index and the Hosoya polynomial of the Jahangir graphs $J_5,m$ for all integer number $m \geq …
Let $G$ be a finite group. The intersection graph of $G$ is a graph whose vertex set is the set of all proper non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K \neq \{e\}$, where $e$ is the identity of the group $G$. In this paper, we investigate some properties and exploring some topological indices such as Wiener, Hyper-Wiener, first and second Zagreb, Schultz, Gutman and eccentric connectivity indices of the intersection graph of $D_{2n}$ for $n=p^2$, $p$ is prime. We also find the metric dimension and the resolving polynomial of the int…
The Wiener index of a graph $W(G)$ is a well studied topological index for graphs. An outstanding problem of Šolt{é}s is to find graphs $G$ such that $W(G)=W(G-v)$ for all vertices $v\in V(G)$, with the only known example being $G=C_{11}$. We relax this problem by defining a notion of Wiener indices for signed graphs, which we denote by $W_σ(G)$, and under this relaxation we construct many signed graphs such that $W_σ(G)=W_σ(G-v)$ for all $v\in V(G)$. This ends up being related to a problem of independent interest, which asks when it is possible to $2$-color the edges of a graph $G$ such that …
The recently developed atom-bond sum-connection (ABS) index is a variation of the well-studied graph-based molecular descriptors connectivity (Randic), atom-bond connectivity, and sum-connectivity indices.We present the minimum ABS index of trees with a given number of pendent vertices in this paper, which provides a solution to the problem proposed by A. Ali, I. Gutman and I. Redzepovic, Atom-Bond Sum-Connectivity Index of Unicyclic Graphs and Some Applications, Electron. J. Math.(2023) 1-7
Many existing degree based topological indices can be classified as bond incident degree (BID) indices, whose general form is $BID(G)=\sum_{uv\in E(G)}$ $Ψ(d_{u},d_{v})$, where $uv$ is the edge connecting the vertices $u,v$ of the graph $G$, $E(G)$ is the edge set of $G$, $d_{u}$ is the degree of the vertex $u$ and $Ψ$ is a non-negative real valued (symmetric) function of $d_{u}$ and $d_{v}$. Here, it has been proven that if the extension of $Ψ$ to the interval $[0,\infty)$ satisfies certain conditions then the extremal $(n,m)$-graph with respect to the BID index (corresponding to $Ψ$) must co…
On comparing Zagreb indices
设$G=(V,E)$是具有$n = |V|$顶点和$m = |E|$边的简单图.第一和第二萨格勒布指数是最古老和最著名的拓扑指数之一，定义为$M_1 = \sum_{i \in V} d_i^2$和$M_2 = \sum_{(i, j) \in E} d_i d_j$，其中$d_i$表示顶点$i$的度。最近提出的猜想$M_1 / n \leqslant M_2 / m$已被证明适用于树、单圈图和化学图，而反例则被发现适用于连通图和不连通图。我们的目标是双重的，既支持一个猜想，也反对它。首先，我们证明了表达式$M_1/n$和$M_2/m$有相同的上下界，这两个上界对正则图是相等的。我们还建立了可变的第一和第二萨格勒布指数的急剧下界。其次，我们证明了对于任何固定的数$k\geqslant 2$，存在一个连通图的$k$圈，其中$M_1/n>M_2/m$成立，有效地表明，猜想不能成立，除非存在某种限制的数量的循环或最大顶点度的图。特别是，我们证明了猜想持有细分图。
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