In this paper, we define the quotinet graphs. In particular, we introduce the boundary quotient graphs, admissible boundary quotient graphs and subgraph boundary qoutient graphs. By the property of the quotient spaces, the boundary quotients need not be invariants on graphs. But, in certain cases, the admissible boundary quotient can be an invariant. The graph index is defined and observed. The concept is somewhat artificial, but we can construct the group-theory-like index on graphs, with respect to the subgraph qoutient graphs.
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.
A vertex-degree-based topological index named as Sombor index of a simple graph G with n vertices was recently introduced by I. Gutman. In this paper, we find Sombor index of m-splitting graph and m-shadow graph. Also, we determine relation between energy and Sombor energy of m-splitting graph and m-shadow graph of k-regular graph.
The Sombor index is a topological index in graph theory defined by Gutman in 2021. In this article we find the maximum Sombor index of unicyclic graphs with a fixed number of pendant vertices. We also provide the unique graph among the chosen class where the maximum Sombor index is attained.
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.
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 …
For a graph $G$ with vertex set $V_{G}$ and edge set $E_{G}$, the symmetric division deg index is defined as $SDD(G)=\sum\limits_{uv\in E_{G}}(\frac{d_{u}}{d_{v}}+\frac{d_{v}}{d_{u}})$, where $d_{u}$ denotes the degree of vertex $u$ in $G$. In 2018, Furtula et al. confirmed the quality of SDD index exceeds that of some more popular VDB indices, in particular that of the GA index. They shown a close connection between the SDD index and the earlier well-established GA index. Thus it is meaningful and important to consider the chemical and mathematical properties of the SDD index. In this paper, …
The concept of geometric-arithmetic index was introduced in the chemical graph theory recently, but it has shown to be useful. The aim of this paper is to obtain new inequalities involving the geometric-arithmetic index $GA_1$ and characterize graphs extremal with respect to them. Besides, we prove inequalities involving the geometric-arithmetic index of line graphs.
In this paper, we derive closed-form formulas for Kirchhoff index and Wiener index of linear pentagonal cylinder graph and linear pentagonal Mobius chain graph. We also obtain explicit formulas for finding total number of spanning trees for both the graphs.
The Wiener index of a connected graph is defined as the sum of the distances between all unordered pair of its vertices. In this paper, we characterize the graphs which extremize the Wiener index among all graphs on $n$ vertices with $k$ pendant vertices. We also characterize the graph which minimizes the Wiener index over the graphs on $n$ vertices with $s$ cut vertices.
Let $G$ be a nonempty simple graph with a vertex set $V(G)$ and an edge set $E(G)$. For every injective vertex labeling $f:V(G)\to\mathbb{Z}$, there are two induced edge labelings, namely $f^+:E(G)\to\mathbb{Z}$ defined by $f^+(uv)=f(u)+f(v)$, and $f^-:E(G)\to\mathbb{Z}$ defined by $f^-(uv)=|f(u)-f(v)|$. The sum index and the difference index are the minimum cardinalities of the ranges of $f^+$ and $f^-$, respectively. We provide upper and lower bounds on the sum index and difference index, and determine the sum index and difference index of various families of graphs. We also provide an inter…
Very recently, a bond-additive topological descriptor, known as the Mostar index, has been proposed as a measure of peripherality in graphs and networks. In this article, we compute the Mostar index of corona product, Cartesian product, join, lexicographic product, Indu-Bala product and subdivision vertex-edge join of graphs and apply these results to find the Mostar index of various classes of chemical graphs and nanostructures.
We show that on cactus graphs the Szeged index is bounded above by twice the Wiener index. For the revised Szeged index the situation is reversed if the graph class is further restricted. Namely, if all blocks of a cactus graph are cycles, then its revised Szeged index is bounded below by twice its Wiener index. Additionally, we show that these bounds are sharp and examine the cases of equality. Along the way, we provide a formulation of the revised Szeged index as a sum over vertices, which proves very helpful, and may be interesting in other contexts.
Recently in 2021, Gutman introduced the Sombor index of a graph, a novel degree-based topological index. It has been shown that the Sombor index efficiently models the thermodynamic properties of chemical compounds. Assume $\mathbb{B}_n^k$ (resp. $\mathbb{V}_n^k$) comprises all graphs with order $n$ having number of bridges (resp. vertex-connectivity) $k$. Horoldagva & Xu (2021) characterized graphs achieving the maximum Sombor index of graphs in $\mathbb{B}_n^k$. This paper characterizes graphs achieving the minimum Sombor index in $\mathbb{B}_n^k$. Certain auxiliary operation on graphs in $\…
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.
The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. A connected graph is Eulerian if its vertex degrees are all even. In [Gutman, Cruz, Rada, Wiener index of Eulerian Graphs, Discrete Applied Mathematics 132 (2014), 247-250] the authors proved that the cycle is the unique graph maximising the Wiener index among all Eulerian graphs of given order. They also conjectured that for Eulerian graphs of order $n \geq 26$ the graph consisting of a cycle on $n-2$ vertices and a triangle that share a vertex is the unique Eulerian graph with second la…
Given an edge-coloring of a graph $G$, we associate to every vertex $v$ of $G$ the set of colors appearing on the edges incident with $v$. The palette index of $G$ is defined as the minimum number of such distinct sets, taken over all possible edge-colorings of $G$. A graph with a small palette index admits an edge-coloring which can be locally considered to be almost symmetric, since few different sets of colors appear around its vertices. Graphs with palette index $1$ are $r$-regular graphs admitting an $r$-edge-coloring, while regular graphs with palette index $2$ do not exist. Here, we cha…
A graph is said to be \textit{$H$-minor free} if it does not contain $H$ as a minor. In this paper, we characteristic the unique extremal graph with maximal $Q$-index among all $n$-vertex $K_{1,t}$-minor free graphs ($t\ge3$).
A mixed graph $\widetilde{G}$ is obtained by orienting some edges of $G$, where $G$ is the underlying graph of $\widetilde{G}$. The positive inertia index, denoted by $p^{+}(G)$, and the negative inertia index, denoted by $n^{-}(G)$, of a mixed graph $\widetilde{G}$ are the integers specifying the numbers of positive and negative eigenvalues of the Hermitian adjacent matrix of $\widetilde{G}$, respectively. In this paper, we study the positive and negative inertia index of the mixed unicyclic graph. Moreover, we give the upper and lower bounds of the positive and negative inertia index of the …
Resistance distance-based graph invariants of subdivisions and triangulations of graphs
我们研究了三种基于电阻距离的图不变量：Kirchhoff指标，以及两种修正，即乘性度-Kirchhoff指标和加性度-Kirchhoff指标。在出版的工作中，本文作者之一（2014）和Sun等人（2014）独立地获得了图的细分的Kirchhoff指数的（不同的）公式。Huang et al.（2014）得到了图的三角剖分的Kirchhoff指数公式。本文首先利用$G$的不变量，导出了剖分和三角剖分的加性度Kirchhoff指标和乘性度Kirchhoff指标的计算公式，以及一个新的三角剖分的Kirchhoff指标计算公式。然后，我们的Kirchhoffian图不变量之间的比较细分和三角剖分。最后给出了图的迭代剖分和三角剖分的图不变量的计算公式。
The index of a signed graph is the largest eigenvalue of its adjacency matrix. For positive integers $n$ and $m\le n^2/4$, we determine the maximal index of complete signed graphs with $n$ vertices and $m$ negative edges. This settles (the corrected version of) a conjecture by Koledin and Stanić (2017).
Došlić et al.~defined the Mostar index of a graph $G$ as $\sum\limits_{uv\in E(G)}|n_G(u,v)-n_G(v,u)|$, where, for an edge $uv$ of $G$, the term $n_G(u,v)$ denotes the number of vertices of $G$ that have a smaller distance in $G$ to $u$ than to $v$. Contributing to conjectures posed by Došlić et al., we show that the Mostar index of bipartite graphs of order $n$ is at most $\frac{\sqrt{3}}{18}n^3$, and that the Mostar index of split graphs of order $n$ is at most $\frac{4}{27}n^3$.
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.
The edge-Wiener index of a connected graph $G$ is defined as the Wiener index of the line graph of $G$. In this paper it is shown that the edge-Wiener index of an edge-weighted graph can be computed in terms of the Wiener index, the edge-Wiener index, and the vertex-edge-Wiener index of weighted quotient graphs which are defined by a partition of the edge set that is coarser than $Θ^*$-partition. Thus, already known analogous methods for computing the edge-Wiener index of benzenoid systems and phenylenes are greatly generalized. Moreover, reduction theorems are developed for the edge-Wiener in…
Let $Γ=(K_{n},H^-)$ be a signed complete graph whose negative edges induce a subgraph $H$. The index of $Γ$ is the largest eigenvalue of its adjacency matrix. In this paper we study the index of $Γ$ when $H$ is a unicyclic graph. We show that among all signed complete graphs of order $n>5$ whose negative edges induce a unicyclic graph of order $k$ and maximizes the index, the negative edges induce a triangle with all remaining vertices being pendant at the same vertex of the triangle.
The Wiener index of a connected graph is the sum of the distances between all pairs of vertices in the graph. It was conjectured that the Wiener index of an $n$-vertex maximal planar graph is at most $\lfloor\frac{1}{18}(n^3+3n^2)\rfloor$. We prove this conjecture and for every $n$, $n \geq 10$, determine the unique $n$-vertex maximal planar graph for which this maximum is attained.
Let $G=(V,E)$ be a finite simple graph. The Sombor index $SO(G)$ of $G$ is defined as $\sum_{uv\in E(G)}\sqrt{d_u^2+d_v^2}$, where $d_u$ is the degree of vertex $u$ in $G$. In this paper, we study this index for certain graphs and we examine the effects on $SO(G)$ when $G$ is modified by operations on vertex and edge of $G$. Also we present bounds for the Sombor index of join and corona product of two graphs.
Fullerenes are molecules in the form of cage-like polyhedra, consisting solely of carbon atoms. Fullerene graphs are mathematical models of fullerene molecules. The transmission of a vertex $v$ of a graph is the sum of distances from $v$ to all the other vertices. The number of different vertex transmissions is called the Wiener complexity of a graph. Some calculation results on the Wiener complexity and the Wiener index of fullerene graphs of order $n \le 216$ are presented. Structure of graphs with the maximal Wiener complexity or the maximal Wiener index is discussed and formulas for the Wi…
