设$G$是一个阶为$n$的连通图。$G$的Wiener指数$W(G)$是$G$的所有无序顶点对之间距离之和。在本文中，我们证明，如果图还包含一个度数较大的顶点，那么关于阶为$n$、最小度为$δ$的图的Wiener指数的著名上界$\big( \frac{n}{δ+1}+2\big) {n \tchoose 2}$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)]可以被显著改进。具体地，我们给出了关于阶为$n$、最小度为$δ$、最大度为$Δ$的图$G$的Wiener指数的渐近尖锐界$W(G) \tleq {n-Δ+δ\tschoose 2} \frac{n+2Δ}{δ+1}+ 2n(n-1)$。我们证明了对于无三角形图也有类似的结果，并确定了给定阶、最小度和最大度的$C_4$-free图的Wiener指数的一个界，并表明它在某种意义上是最好可能的。
英文摘要：
Let $G$ be a connected graph of order $n$.The Wiener index $W(G)$ of $G$ is the sum of the distances between all unordered pairs of vertices of $G$. In this paper we show that the well-known upper bound $\big( \frac{n}{δ+1}+2\big) {n \choose 2}$ on the Wiener index of a graph of order $n$ and minimum degree $δ$ [M. Kouider, P. Winkler, Mean distance and minimum degree. J. Graph Theory 25 no. 1 (1997)] can be improved significantly if the graph contains also a vertex of large degree. Specifically, we give the asymptotically sharp bound $W(G) \leq {n-Δ+δ\choose 2} \frac{n+2Δ}{δ+1}+ 2n(n-1)$ on the Wiener index of a graph $G$ of order $n$, minimum degree $δ$ and maximum degree $Δ$. We prove a similar result for triangle-free graphs, and we determine a bound on the Wiener index of $C_4$-free graphs of given order, minimum and maximum degree and show that it is, in some sense, best possible.
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…
In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. We introduce a generalized Wiener polarity index $W_k (G)$ as the number of unordered pairs of vertices ${u, v}$ of $G$ such that the shortest distance $d (u, v)$ between $u$ and $v$ is $k$ (this is actually the $k$-th coefficient in the Wiener polynomial). For $k = 3$, we get standard Wiener polarity index. Furthermore, we generalize the terminal Wiener index $TW_k (G)$ as the sum of distances between all pairs of vertic…
Let $S$ be a set of vertices of a connected graph $G$. The Steiner distance of $S$ is the minimum size of a connected subgraph of $G$ containing all the vertices of $S$. The sum of all Steiner distances on sets of size $k$ is called the Steiner $k$-Wiener index, hence for $k=2$ we get the Wiener index. The modular graphs are graphs in which every three vertices $x, y$ and $z$ have at least one median vertex $m(x,y,z)$ that belongs to shortest paths between each pair of $x, y$ and $z$. The Steiner 3-Wiener index of a modular graph is expressed in terms of its Wiener index. As a corollary formul…
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…
In this paper, we discuss the Hamiltonicity of graphs in terms of Wiener index, hyper-Wiener index and Harary index of their quasi-complement or complement. Firstly, we give some sufficient conditions for an balanced bipartite graph with given the minimum degree to be traceable and Hamiltonian, respectively. Secondly, we present some sufficient conditions for a nearly balanced bipartite graph with given the minimum degree to be traceable. Thirdly, we establish some conditions for a graph with given the minimum degree to be traceable and Hamiltonian, respectively. Finally, we provide some condi…
In this paper, we study the Steiner hyper-Wiener index of a graph, which is obtained from the standard hyper-Wiener index by replacing the classical graph distance with the Steiner distance. It is shown how this index is related to the Steiner Hosoya polynomial, which generalizes similar result for the standard hyper-Wiener index. Next, we show how the Steiner $3$-hyper-Wiener index of a modular graph can be expressed by using the classical graph distances. As the main result, a method for computing this index for median graphs is developed. Our method makes computation of the Steiner $3$-hype…
The Wiener index of a (hyper)graph is calculated by summing up the distances between all pairs of vertices. We determine the maximum possible Wiener index of a connected $n$-vertex $k$-uniform hypergraph and characterize for every~$n$ all hypergraphs attaining the maximum Wiener index.
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…
The Wiener index of a connected graph is the sum of topological distances between all pairs of vertices. Since Wang gave a mistake result on the maximum Wiener index for given tree degree sequence, in this paper, we investigate the maximum Wiener index of trees with given degree sequences and extremal trees which attain the maximum value.
The Wiener index of a connected graph is the sum of distances between all unordered pairs of vertices. Sam Spiro [The Wiener index of signed graphs, Appl. Math. Comput., 416(2022)126755] recently introduced the Wiener index for a signed graph and conjectured that the path $P_n$ with alternating signs has the minimum Wiener index among all signed trees with $n$ vertices. By constructing an infinite family of counterexamples, we prove that the conjecture is false whenever $n$ is at least 30.
Hyper-Wiener index was introduced as one of the main generalizations of the well known Wiener index. Through the years properties of the Wiener index have been extensively studied in both Mathematics and Chemistry. The Hyper-Wiener index, although received much attention, is far from being thoroughly examined due to its complex definition. We consider the local version of the Hyper-Wiener index ($WW(G)$), defined as $ww_G(v)=\sum\limits_{u\in V(G)}(d^2(u,v)+d(u,v))$ for a vertex $v$ in a graph $G$, in trees. For established results on the Wiener index ($W(.)$), we present analogous studies on …
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.
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 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.
The Wiener index of a graph is the sum of the distances between all pairs of vertices, it has been one of the main descriptors that correlate achemical compound's molecular graph with experimentally gathered data regarding the compound's characteristics. The tree that minimizes the Wiener index among trees of given maximal degree was studied. We characterize trees that achieve the maximum and minimum Wiener index, given the number of vertices and the degree sequence.
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.
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…
The Wiener polarity index of a graph G is the number of unordered pairs of vertices u, v such that the distance between u and v is 3. In this paper we give an explicit formula for the Wiener polarity index of cactus graphs. We also deduce formulas for some special cactus graphs.
The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide formulae for the minimum Wiener index of simple triangulations and quadrangulations with connectivity at least $c$, and provide the extremal structures, which attain those values. Our main tool is setting upper bounds for the maximum degree in highly connected triangulations and quadrangulations.
In this paper, we investigate how the Wiener index of unicyclic graphs varies with graph operations. These results are used to present a sharp lower bound for the Wiener index of unicyclic graphs of order $n$ with girth and the matching number $β\ge \frac{3g}{2}$. Moreover, we characterize all extremal graphs which attain the lower bound.
The edge-hyper-Wiener index of a connected graph $G$ is defined as $WW_e(G) = \frac{1}{2}\sum_{\lbrace e,f\rbrace \subseteq E(G)}d(e,f) + \frac{1}{2}\sum_{\lbrace e,f\rbrace \subseteq E(G)}d(e,f)^2$. We develop a method for computing the edge-hyper-Wiener index of partial cubes, which constitute a large class of graphs with a lot of applications. It is also shown how the method can be applied to trees. Furthermore, an algorithm for computing the edge-hyper-Wiener index of benzenoid systems is obtained. Finally, the algorithm is used to correct already known closed formulas for the edge-Wiener …
We establish several contraction formulas for Kirchhoff index. We relate Kirchhoff index with some other metrized graph invariants. By applying our contraction formulas successively when the graph is a tree, we derive new formulas for Wiener index and obtain some previously known Wiener index formulas with new proofs.
Let $S$ be a set of vertices of a connected graph $G$. The Steiner distance of $S$ is the minimum size of a connected subgraph of $G$ containing all the vertices of $S$. The Steiner $k$-Wiener index is the sum of all Steiner distances on sets of $k$ vertices of $G$. Different simple methods for calculating the Steiner $k$-Wiener index of block graphs are presented.
The Wiener index of a graph, which is the sum of the distances between all pairs of vertices, has been well studied. Recently, Sills and Wang in 2012 proposed two conjectures on the maximal Wiener index of trees with a given degree sequence. This note proves one of the two conjectures and disproves the other.
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…
We prove that the Wiener Index $W(G)$ of a Maximal Planar graph $G$ with $n$ vertices satisfies $W(G) \leq \Big{\lfloor} \frac{1}{18}(n^3 + 3n^2) \Big{\rfloor}$ for $3 \leq n \leq 18$.
The Wiener index of a strong digraph $D$ is defined as the sum of the distances between all ordered pairs of vertices. This definition has been extended to digraphs that are not necessarily strong by defining the distance from a vertex $a$ to a vertex $b$ as $0$ if there is no path from $a$ to $b$ in $D$. Knor, uSkrekovski and Tepeh [Some remarks on Wiener index of oriented graphs. Appl.\ Math.\ Comput.\ {\bf 273}] considered orientations of graphs with maximum Wiener index. The authors conjectured that for a given tree $T$, an orientation $D$ of $T$ of maximum Wiener index always contains a v…
The Wiener index is maximized over the set of trees with the given vertex weight and degree sequences. This model covers the traditional "unweighed" Wiener index, the terminal Wiener index, and the vertex distance index. It is shown that there exists an optimal caterpillar. If weights of internal vertices increase in their degrees, then an optimal caterpillar exists with weights of internal vertices on its backbone monotonously increasing from some central point to the ends of the backbone, and the same is true for pendent vertices. A tight upper bound of the Wiener index value is proposed and…
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.
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 terminal Wiener index of a tree is the sum of distances for all pairs of pendent vertices, which recently arises in the study of phylogenetic tree reconstruction and the neighborhood of trees. This paper presents a sharp upper and lower bounds for the terminal Wiener index in terms of its order and diameter and characterizes all extremal trees which attain these bounds. In addition, we investigate the properties of extremal trees which attain the maximum terminal Wiener index among all trees of order $n$ with fixed maximum degree.
In 1997 Klavžar and Gutman suggested a generalization of the Wiener index to vertex-weighted graphs. We minimize the Wiener index over the set of trees with the given vertex weights' and degrees' sequences and show an optimal tree to be the, so-called, Huffman tree built in a bottom-up manner by sequentially connecting vertices of the least weights.
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…
In this paper, we prove tight sufficient conditions for traceability and Hamiltonicity of connected graphs with given minimum degree, in terms of Wiener index and Harary index. We also prove some result on Hamiltonicity of balanced bipartite graphs in the similar fashion. In two recent papers \cite{LDJ2016,LDJ2017}, Liu et al. corrected some previous work on traceability of connected graphs in terms of Wiener index and Harary index, respectively, such as \cite{HW2013,Y2013}. We generalize these results and give short and unified proofs. All results in this paper are best possible.
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.
In this note, we discuss the method explained in the recent paper [P. Manuel, I. Rajasingh, B. Rajan, R. Sundara Rajan, A New Approach To Compute Wiener Index, Journal of Computational and Theoretical Nanoscience 10, (2013) 1515-1521.] for computing the Wiener index of special chemical graphs. The method is actually already well-known and equivalent to the 'cut method' introduced in 1995 by Klavzar, Gutman and Mohar, and used in multiple papers for computing various distance based graph invariants.
The Wiener index of a connected graph is the summation of all distances between unordered pairs of vertices of the graph. In this paper, we give an upper bound on the Wiener index of a $k$-connected graph $G$ of order $n$ for integers $n-1>k \ge 1$: \[W(G) \le \frac{1}{4} n \lfloor \frac{n+k-2}{k} \rfloor (2n+k-2-k\lfloor \frac{n+k-2}{k} \rfloor).\] Moreover, we show that this upper bound is sharp when $k \ge 2$ is even, and can be obtained by the Wiener index of Harary graph $H_{k,n}$.
A formula based on a vertex contributions of the Steiner $k$-Wiener index is induced by a newly introduced $k$-Steiner betweenness centrality, which measures the number of $k$-Steiner trees that include a particular vertex as a non-terminal vertex. This generalizes Škrekovski and Gutman's Vertex version of the Wiener Theorem and a result of Gago on the average betweenness centrality and the average distance in general graphs.
The Steiner distance of vertices in a set $S$ is the minimum size of a connected subgraph that contain these vertices. The sum of the Steiner distances over all sets $S$ of cardinality $k$ is called the Steiner $k$-Wiener index and studied as the natural generalization of the famous Wiener index in chemical graph theory. In this paper we study the extremal structures, among trees with a given segment sequence, that maximize or minimize the Steiner $k$-Wiener index. The same extremal problems are also considered for trees with a given number of segments.
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.
The Wiener polarity index of a graph is defined as the number of unordered pairs of vertices at distance three. In recent years, this topological index was extensively studied since it has many known applications in chemistry and also in network theory. In this paper, we generalize the result of Behmaram, Yousefi-Azari, and Ashrafi proved in 2012 for calculating the Wiener polarity index of a graph. An important advantage of our generalization is that it can be used for graphs that contain $4$-cycles and also for graphs whose different cycles have more than one common edge. In addition, using …
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}$.
In this paper, we consider a molecular descriptor called the Wiener polarity index, which is defined as the number of unordered pairs of vertices at distance three in a graph. Molecular descriptors play a fundamental role in chemistry, materials engineering, and in drug design since they can be correlated with a large number of physico-chemical properties of molecules. As the main result, we develop a method for computing the Wiener polarity index for two basic and most commonly studied families of molecular graphs, benzenoid systems and carbon nanotubes. The obtained method is then used to fi…
The Wiener index (i.e., the total distance or the transmission number), defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. In this article we summarize some results, conjectures and problems on this molecular descriptor, with emphasis on works we were involved in.
The eccentricity of a vertex $u$ in a connected graph $G$ is the distance between $u$ and a vertex farthest from it; the eccentric sequence of $G$ is the nondecreasing sequence of the eccentricities of $G$. In this paper, we determine the unique tree that minimises the Wiener index, i.e. the sum of distances between all unordered vertex pairs, among all trees with a given eccentric sequence. We show that the same tree maximises the number of subtrees among all trees with a given eccentric sequence, thus providing another example of negative correlation between the number of subtrees and the Wi…
The edge-Wiener index $W_e(G)$ of a connected graph $G$ is the sum of distances between all pairs of edges of $G$. A connected graph $G$ is said to be a cactus if each of its blocks is either a cycle or an edge. Let $\mathcal{G}_{n,t}$ denote the class of all cacti with $n$ vertices and $t$ cycles. In this paper, the upper bound and lower bound on the edge-Wiener index of graphs in $\mathcal{G}_{n,t}$ are identified and the corresponding extremal graphs are characterized.
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…
This paper is devoted to Wiener index of directed graphs, more precisely of directed grids. The grid $G_{m,n}$ is the Cartesian product $P_m\Box P_n$ of paths on $m$ and $n$ vertices, and in a particular case when $m=2$, it is a called the ladder graph $L_n$. Kraner Šumenjak et al. proved that the maximum Wiener index of a digraph, which is obtained by orienting the edges of $L_n$, is obtained when all layers isomorphic to one factor are directed paths directed in the same way except one (corresponding to an endvertex of the other factor) which is a directed path directed in the opposite way. …
The cut method has been proved to be extremely useful in chemical graph theory. In this paper the cut method is extended to hypergraphs. More precisely, the method is developed for the Wiener index of $k$-uniform partial cube-hypergraphs. The method is applied to cube-hypergraphs and hypertrees. Extensions of the method to hypergraphs arising in chemistry which are not necessary $k$-uniform and/or not necessary linear are also developed.
We prove a conjecture of Nadjafi-Arani, Khodashenas and Ashrafi on the difference between the Szeged and Wiener index of a graph. Namely, if $G$ is a 2-connected non-complete graph on $n$ vertices, then $Sz(G)-W(G)\ge 2n-6$. Furthermore, the equality is obtained if and only if $G$ is the complete graph $K_{n-1}$ with an extra vertex attached to either $2$ or $n-2$ vertices of $K_{n-1}$. We apply our method to strengthen some known results on the difference between the Szeged and Wiener index of bipartite graphs, graphs of girth at least five, and the difference between the revised Szeged and W…
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 the computing one of topological indices of One-pentagonal carbon Nanocone. One-pentagonal carbon nanocone consists of one pentagone as its core surrounded by layers of hexagons .if there are n layers,then the graph of this molecules is denoted by G_n .In This paper our aim is to calculate the hyper-Wiener index of G_n explicitly .
We resolve two conjectures of Hriňáková, Knor and Škrekovski (2019) concerning the relationship between the variable Wiener index and variable Szeged index for a connected, non-complete graph, one of which would imply the other. The strong conjecture is that for any such graph there is a critical exponent in $(0,1]$, below which the variable Wiener index is larger and above which the variable Szeged index is larger. The weak conjecture is that the variable Szeged index is always larger for any exponent exceeding $1$. They proved the weak conjecture for bipartite graphs, and the strong conjectu…
