A. Jahanbani, L. Shahbazi, S. M. Sheikholeslami, R. Rasi, J. Rodriguez
The forgotten topological index of a graph $G$, denoted by $F(G)$, is defined as the sum of weights $d(u)^{2}+d(v)^{2}$ over all edges $uv$ of $G$ , where $d(u)$ denotes the degree of a vertex $u$. In this paper, we give sharp upper bounds of the F-index (forgotten topological index) over bicyclic graphs, in terms of the order and maximum degree.
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The harmonic index of a graph $G$, is defined as the sum of weights $\frac{2}{d(u)+d(v)}$ of all edges $uv$ of $G$, where $d(u)$ is the degree of the vertex $u$ in $G$. In this paper we find the minimum harmonic index of bicyclic graph of order $n$ and diameter $d$. We also characterized all bicyclic graphs reaching the minimum bound.
On Sum--Connectivity Index of Bicyclic Graphs
本文确定了具有$n$顶点和匹配数$m$的双圈图的最小和连通度指数，其中$2\le m\le \lfloor\frac{n}{2}\rfloor$是具有$n\ge 5$顶点的双圈图的最小和连通度指数和次最小和连通度指数，以及最大和连通度指数和次最大和连通度指数.给出了极图的特征。
In this paper, we focus on the index ( largest eigenvalue) of the adjacency matrix of connected signed graphs. We give some general results on the index when the corresponding signed graph is perturbed. As applications, we determine the first five largest index among all unbalanced bicyclic graphs on n >= 36 vertices together with the corresponding extremal signed graphs whose index attain these values.
Maximum Estrada Index of Bicyclic Graphs
设$G$是一个阶为$n$的单图，$λ_1(G),λ_2(G),...,λ_n(G)$是$G$的邻接矩阵的特征值. $G$的Esrada指数定义为$EE(G)=\sum_{i=1}^{n}e^{λ_i(G)}$。本文确定了固定阶双圈图中唯一具有最大Estrada指数的图。
On bicyclic graphs with maximal Graovac-Ghorbani index
针对D. Pacheco等人提出的双圈图Graovac-Ghorbani指数上下界猜想，本文确定了双圈图的最大Graovac-Ghorbani指数及对应极图，解决了相关猜想。
The maximum index of signed complete graphs whose negative edges induce a bicyclic graph
本文研究负边诱导双圈图的有符号完全图的最大指数，并确定了使指数达到最大时双圈图的具体结构。
The weighted adjacency matrix $A_{f}(G)$ of a simple graph $G=(V,E)$ is the $|V|\times|V|$ matrix whose $ij$-entry equals $f(d_{i},d_j)$, where $f(x,y)$ is a symmetric function such that $f(d_i,d_j)>0$ if $ij\in E$ and $f(d_i,d_j)=0$ if $ij\notin E$ and $d_i$ is the degree of the vertex $i$. In this paper, we determine the unique graph having the largest spectral radius of $A_{f}(G)$ among all the bicyclic graphs under the assumption that $f(x,y)$ is increasing and convex in $x$ and $f(x_1,y_1)\geq f(x_2,y_2)$ when $|x_1-y_1|>|x_2-y_2|$ and $x_1+y_1=x_2+y_2$. Moreover, we determine the unique …
The zeroth-order general Randić index $R^{0}_{a}$ of a digraph $D$ is the sum of $(d^{+}_{v})^{a}+(d^{-}_{w})^{a}$ over all arcs $vw$ of $D$, where $a$, $d^{+}_{v}$ and $d^{-}_{w}$ are an arbitrary real number, the out-degree of the vertex $v$ and the in-degree of the vertex $w$, respectively. We determine maximum zeroth-order general Randić index of oriented trees, unicyclic and bicyclic graphs in terms of matching number and order in this paper.
The extremal problems on the inertia of weighted bicyclic graphs
设$G_w$是一个赋权图。$G_w$谱中正、负和零特征值的个数称为正惯性指数、负惯性指数和零度，分别用$i_{+}(G_w)$、$i_{-}(G_w)$、$i_{0}(G_w)$表示.在本文中，我们得到了正（分别为正）的精确下界.得到了具有悬挂点的$n$阶加权双圈图的（负）惯性指数.此外，还分别给出了至多有两个正特征值、两个负特征值和至少有$n-4$零特征值的$n$阶加权双圈图的所有特征值.
Bicyclic graphs with maximal revised Szeged index
修正的Szeged指数$Sz^*(G)$被定义为$Sz^*(G)=\sum_{e=uv \in E}(n_u(e)+ n_0(e)/2)(n_v(e)+ n_0(e)/2),$，其中$n_u(e)$和$n_v(e)$分别是$G$中距离顶点$u$比距离顶点$v$更近的顶点数和距离顶点$v$比距离顶点$u$更近的顶点数，而$n_0(e)$是距离$u$和$v$等距离的顶点数。Hansen利用AutoGraphiX对$n \geq 6$阶连通双圈图$G$的修正Szeged指数做出了如下猜想：$$ Sz^*(G)\leq \{{array}{ll} (n^3+n^2-n-1)/4,& {if $n$ is odd}, (n^3+n^2-n)/4, & {if $n$ is even}. {array}. $$具有相等性当且仅当$G$是通过复制一个顶点从圈$C_{n-1}$获得的图。本文对这一猜想给出了一个证实性的证明。
Extremal Zagreb indices of bicyclic hypergraphs
针对线性双圈一致超图，本文刻画了使其Zagreb指数分别达到最大值和最小值的超图结构。
On the Estrada index of unicyclic and bicyclic signed graphs
本文研究符号图Estrada指标的极值问题，刻画了具有最大Estrada指标的单圈符号图，并在具有配对性质的不平衡图类以及不平衡完全二部符号图中确定了取得最大Estrada指标的图。
The First Zagreb Index, the Forgotten Topological Index, the Inverse Degree and Some Hamiltonian Properties of Graphs
本文研究了最小度至少为1的图，给出了第一Zagreb指标的上界及被遗忘的拓扑指标与逆度的下界，并提出了涉及这些指标的充分条件以确保图的哈密顿性质。
Complex unit gain bicyclic graphs with rank 2, 3 or 4
$\mathbb{T}$增益图是由图$G=(V,E)$、圆组$\mathbb{T}=\{z\in C: |z|=1\}$和增益函数$φ:\overrightarrow{E}\rightarrow \mathbb{T}$组成的三重$Φ=(G,\mathbb{T},φ)$，使得$φ(e_{ij})=φ(e_{ji})^{-1}=\overline{φ(e_{ji})}$。用$r(Φ)$表示的$\mathbb{T}$ -增益图$Φ$的秩是$Φ$的邻接矩阵的秩。2015年，Yu，Qu和Tu [ G. H. Yu，H. Qu，J.H.涂，复单位增益图的惯性，应用数学计算。265（2015）619--629 ]获得了$\mathbb{T}$ -增益图的惯性的一些性质。他们刻画了具有小正指数或小负指数的$\mathbb{T}$ -增益单圈图。受此启发，本文刻画了秩为2，3，4的复单位增益双圈图。
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…
F-index of graphs based on new operations related to the join of graphs
图的遗忘拓扑指数或F-指数定义为图的所有顶点的度立方之和。这个指数是由古特曼和特里纳杰斯蒂奇在40多年前提出的。本文给出了与图的并有关的剖分图的新运算的F-指标。
In this paper, four novel topological indices named as neighbourhood version of forgotten topological index (F_N), modified neighbourhood version of Forgotten topological index ($F_N^*$), neighbourhood version of second Zagreb index ($M_2^*$)and neighbourhood version of hyper Zagreb index ($HM_N$) are introduced. Here the relatively study depends on the structure-property regression analysis is made to test and compute the chemical applicability of these indices for the prediction of physicochemical properties of octane isomers. Also it is shown that these newly presented indices have well deg…
A topological index is a real number which is derived from a network or a graph by mathematically that characterizes the whole of its structural properties. Recently, there are various topological indices that have been introduced in mathematical chemistry to predict the properties of molecular topology. Among, the degree based topological indices such as Zagreb indices, forgotten topological index, redefined Zagreb index, Randic index, general first Zagreb index, symmetric division deg index and hence so forth are most important, because of their chemical significance. In this work, we study …
F-index of graphs based on four operations related to the lexicographic product
图的遗忘拓扑指数或F-指数被定义为图的所有顶点的度的立方之和。本文研究了Sarala等人[D. Sarala，H. Deng，S.K. Ayyaswamya和S. Balachandrana，基于与词典产品相关的四个新操作的图的萨格勒布指数，\textit{Applied Mathematics and Computation}，309（2017）156- 169。
F-Index of Some Graph Operations
图的F-指数定义为图的顶点度的立方之和。这是在1972年引入的，在同一篇论文中，第一和第二萨格勒布指数被引入来研究总$π$ -电子能量的结构依赖性。但直到那时，这种拓扑指数才得到进一步的研究。最近，Furtula和Gutman [B。富尔图拉岛Gutman，一种被遗忘的拓扑指数，J. Math. Chem.，53（4）（2015）1184- 1190.]重新研究了该指数，并将其命名为“遗忘拓扑指数”或“F-指数”。在该论文中，他们介绍了该指数的一些基本性质，并表明该指数可以增强萨格勒布指数的物理化学适用性。在这里，我们研究了这个指数在几种图形操作下的行为，并将我们的结果应用于寻找不同化学有趣的分子图和纳米结构的F指数。
In a connected graph G, the distance between two vertices of G is the length of a shortest path between these vertices. The eccentricity of a vertex u in G is the largest distance between u and any other vertex of G. The total-eccentricity index τ(G) is the sum of eccentricities of all vertices of G. In this paper, we find extremal trees, unicyclic and bicyclic graphs with respect to total-eccentricity index. Moreover, we find extremal conjugated trees with respect to total-eccentricity index.
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.
On the positive and negative inertia of weighted graphs
（边）赋权图$G$的谱中正、负和零特征值的个数称为赋权图$G$的正惯性指数、负惯性指数和零度，分别用$i_+(G)$、$i_-(G)$、$i_0(G)$表示.本文讨论了赋权树、赋权单圈图和赋权双圈图的正惯性指数和负惯性指数，给出了它们的计算方法。
We present the bounds in terms of other important graph parameters for general Sombor index which generalises both the forgotten index and the Sombor index. We also explore the Nordhaus-Gaddum-type result for the general Sombor index. We present further the relations between general Sombor index and other generalised indices: general Randić index and general sum-connectivity index.
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 $$SO_α(G)=\sum_{uv\in E} \left[d(u)^2+d(v)^2\right]^α$$ where $SO(G)=SO_{0.5}(G)$ is the recently invented Sombor index. In this paper, we show that in the class of connected graphs with a fixed degree sequence (for which the minimum degree being equal to one), there exists a special extremal $BFS$-graph with minimum general Sombor index for $0<α<1$ (resp. maximum general Sombor index for either $α>1$ or $α<0$). Moreover, for any given tree, unicyclic, and bi…
New boundes for Sombor index of Graphs
本研究利用三角不等式、多种拓扑指数及中心性离散性指数推导了图Sombor指数的新界，可给出该指数的估计值与不等式形式的误差区间，明确其精度极限。
The aim of this paper is to obtain new sharp inequalities for a large family of topological indices, including the first variable Zagreb index $M_1^α$, and to characterize the set of extremal graphs with respect to them. Our main results provide lower bounds on this family of topological indices involving just the minimum and the maximum degree of the graph. These inequalities are new even for the first Zagreb, the inverse and the forgotten indices.
The Second Zagreb Indices of Graphs with Given Degree Sequences
图G的第二Zagreb指标记为$M_2(G)=\sum_{uv\in E(G)}d(u)d(v)$。本文研究了具有最大第二Zagreb指标的极图在给定图序列，特别是图双圈序列下的性质。此外，我们还得到了不同度序列的极图的第二Zagreb指标之间的关系。
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。
Motivated by the recently introduced topological index, the Somber index, we define a new topological index of a graph in this paper, we call it Sombor coindex. The Sombor coindex is defined by considering analogous contributions from the pairs of non-adjacent vertices, capturing, thus, and quantifying a possible influence of remote pairs of vertices. We give several properties of the Somber coindex and its relations to the Sombor index, the Zagreb (co)indices, forgotten coindex and other important graph parameters. We also compute the bounds of the Somber coindex of some graph operations and …
Let $G$ be a simple connected graph with the vertex set $V(G)$ and $d_{B}^2(u,v)$ be the biharmonic distance between two vertices $u$ and $v$ in $G$. The biharmonic index $BH(G)$ of $G$ is defined as $$BH(G)=\frac{1}{2}\sum\limits_{u\in V(G)}\sum\limits_{v\in V(G)}d_{B}^2(u,v)=n\sum\limits_{i=2}^{n}\frac{1}{λ_i^2(G)},$$ where $λ_i(G)$ is the $i$-th smallest eigenvalue of the Laplacian matrix of $G$ with $n$ vertices. In this paper, we provide the mathematical relationships between the biharmonic index and some classic topological indices: the first Zagreb index, the forgotten topological index…
Disproof of a conjecture on the edge Mostar index
本文研究连通图的边 Mostar 指数，通过确定双圈图上的严格上界及其极值图，反证了 Liu 等人于 2020 年提出的相关猜想。
The signature of line graphs and power trees
设$G$是一个图，$A(G)$是$G$的邻接矩阵。$G$的签名$s(G)$是$A(G)$的正惯性指数和负惯性指数之差。Ma等人[Positive and negative inertia index of a graph，Linear Algebra and its Applications 438（2013）331-341]证明了$-c_3(G)\leq s(G)\leq c_5(G),$，其中$c_3(G)$和$c_5(G)$分别表示$G$中对于某些整数$k \ge 0$具有长度$4k+3$和$4k+5$的圈数，并证明了该猜想对于树、单环或双环图成立。从已有的结果可知，如果$G$是二部图，则$s(G)=0$是二部图，并且签名与图的奇圈或非二部性密切相关.在本文中，我们证明了猜想成立的线图和权力树。
On the Extremal Graphs with Respect to Bond Incident Degree Indices
许多现有的基于度的拓扑指数可以被分类为键关联度（BID）指数，其一般形式是$BID(G)=\sum_{uv\in E(G)}$ $Ψ(d_{u},d_{v})$，其中$uv$是连接图$G$的顶点$u,v$的边，$E(G)$是$G$的边集，$d_{u}$是顶点$u$的度，并且$Ψ$是$d_{u}$和$d_{v}$的非负实值（对称）函数。这里，它已被证明，如果$Ψ$的扩展到区间$[0,\infty)$满足一定的条件，那么极值$(n,m)$ -图的BID指数（对应于$Ψ$）必须包含至少一个顶点度$n-1$。证明了这些条件对于一般的和连通性指数（其特殊情况是：第一Zagreb指数和Hyper Zagreb指数）、一般的Platt指数（其特殊情况是：第一重新表述的Zagreb指数和Platt指数）和可变的和exdeg指数都是满足的。应用上述结果，在树图、单圈图、双环图、三环图和四环图中刻画了具有上述BID指标最大值的图。其中一些结果是新的和已经存在的结果证明在一个更短和更统一的方式。
On Hyperbolic Sombor index of graphs
本文纠正了Hyperbolic Sombor指标的先前错误，为树、单圈图和双圈图建立了新结果，推导了关于边数和度数的上下界并刻画了极值图。
Topological indices are graph invariants numeric quantities, which are utilized by researchers to analyze a variety of physiochemical aspects of molecules. The goal of developing topological indices is to give each chemical structure a numerical value while maintaining the highest level of differentiation. Using these indices, the classification of various structures, and their physiochemical and biological properties can be predicted. In this paper, the leap and leap hyper Zagreb indices, as well as their polynomials for a zigzag benzenoid system $Z_{p}$ and a rhombic benzenoid system $R_{p}$…
On Topological Indices in Trees: Fibonacci Degree Sequences and Bounds
该研究针对树的拓扑指数，推导斐波那契度序列树的Albertson指数精确公式，得到不同树结构下该指数的极值界，并关联第二Zagreb指数与遗忘指数建立新上界。
For a graph $G=(V,E)$ and $v_{i}\in V$, denote by $d_{i}$ the degree of vertex $v_{i}$. Let $f(x, y)>0$ be a real symmetric function in $x$ and $y$. The weighted adjacency matrix $A_{f}(G)$ of a graph $G$ is a square matrix, where the $(i,j)$-entry is equal to $\displaystyle f(d_{i}, d_{j})$ if the vertices $v_{i}$ and $v_{j}$ are adjacent and 0 otherwise. Li and Wang \cite{U9} tried to unify methods to study spectral radius of weighted adjacency matrices of graphs weighted by various topological indices. If $\displaystyle f'_{x}(x, y)\geq0$ and $\displaystyle f''_{x}(x, y)\geq0$, then $\displ…
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