The second-minimum Wiener index of cacti with given cycles

2020 ◽  
pp. 2150039
Author(s):  
Hanyuan Deng ◽  
G. C. Keerthi Vasan ◽  
S. Balachandran
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
Index Formula ◽  
Extremal Graphs ◽  
Unique Graph ◽  
Sum Of Distances ◽  
Appl Math

The Wiener index [Formula: see text] of a connected graph [Formula: see text] is the sum of distances between all pairs of vertices of [Formula: see text]. A connected graph [Formula: see text] is said to be a cactus if each of its blocks is either a cycle or an edge. Let [Formula: see text] be the set of all [Formula: see text]-vertex cacti containing exactly [Formula: see text] cycles. Liu and Lu (2007) determined the unique graph in [Formula: see text] with the minimum Wiener index. Gutman, Li and Wei (2017) determined the unique graph in [Formula: see text] with maximum Wiener index. In this paper, we present the second-minimum Wiener index of graphs in [Formula: see text] and identify the corresponding extremal graphs, which solve partially the problem proposed by Gutman et al. [Cacti with [Formula: see text]-vertices and [Formula: see text] cycles having extremal Wiener index, Discrete Appl. Math. 232 (2017) 189–200] in 2017.

Wiener Index of k-Connected Graphs

2021 ◽  
pp. 2142005
Author(s):  
Xiang Qin ◽  
Yanhua Zhao ◽  
Baoyindureng Wu
Keyword(s):  
Positive Integer ◽  
Connected Graph ◽  
Wiener Index ◽  
Index Formula ◽  
Connected Graphs ◽  
Sum Of Distances

The Wiener index [Formula: see text] of a connected graph [Formula: see text] is the sum of distances of all pairs of vertices in [Formula: see text]. In this paper, we show that for any even positive integer [Formula: see text], and [Formula: see text], if [Formula: see text] is a [Formula: see text]-connected graph of order [Formula: see text], then [Formula: see text], where [Formula: see text] is the [Formula: see text]th power of a graph [Formula: see text]. This partially answers an old problem of Gutman and Zhang.

scholarly journals On the Generalized Distance Energy of Graphs

Mathematics ◽  
2019 ◽  
Vol 8 (1) ◽  
pp. 17 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Hilal A. Ganie ◽  
Yilun Shang
Keyword(s):  
Lower Bounds ◽  
Complete Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Wiener Index ◽  
Diagonal Matrix ◽  
Upper And Lower Bounds ◽  
Star Graph ◽  
Extremal Graphs ◽  
Generalized Distance

The generalized distance matrix D α ( G ) of a connected graph G is defined as D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where 0 ≤ α ≤ 1 , D ( G ) is the distance matrix and T r ( G ) is the diagonal matrix of the node transmissions. In this paper, we extend the concept of energy to the generalized distance matrix and define the generalized distance energy E D α ( G ) . Some new upper and lower bounds for the generalized distance energy E D α ( G ) of G are established based on parameters including the Wiener index W ( G ) and the transmission degrees. Extremal graphs attaining these bounds are identified. It is found that the complete graph has the minimum generalized distance energy among all connected graphs, while the minimum is attained by the star graph among trees of order n.

scholarly journals Joins, coronas and their vertex-edge Wiener polynomials

2016 ◽  
Vol 47 (2) ◽  
pp. 163-178
Author(s):  
Mahdieh Azari ◽  
Ali Iranmanesh
Keyword(s):  
Generating Function ◽  
Connected Graph ◽  
Wiener Index ◽  
First Derivative ◽  
Sum Of Distances ◽  
Simple Connected Graph ◽  
Product Of Graphs ◽  
Corona Product

The vertex-edge Wiener index of a simple connected graph $G$ is defined as the sum of distances between vertices and edges of $G$. The vertex-edge Wiener polynomial of $G$ is a generating function whose first derivative is a $q-$analog of the vertex-edge Wiener index. Two possible distances $D_1(u, e|G)$ and $D_2(u, e|G)$ between a vertex $u$ and an edge $e$ of $G$ can be considered and corresponding to them, the first and second vertex-edge Wiener indices of $G$, and the first and second vertex-edge Wiener polynomials of $G$ are introduced. In this paper, we study the behavior of these indices and polynomials under the join and corona product of graphs. Results are applied for some classes of graphs such as suspensions, bottlenecks, and thorny graphs.

scholarly journals Hyper-Wiener indices of polyphenyl chains and polyphenyl spiders

2019 ◽  
Vol 17 (1) ◽  
pp. 668-676
Author(s):  
Tingzeng Wu ◽  
Huazhong Lü
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Wiener Index ◽  
Upper And Lower Bounds ◽  
Extremal Graphs ◽  
Recurrence Formulae

Abstract Let G be a connected graph and u and v two vertices of G. The hyper-Wiener index of graph G is $\begin{array}{} WW(G)=\frac{1}{2}\sum\limits_{u,v\in V(G)}(d_{G}(u,v)+d^{2}_{G}(u,v)) \end{array}$, where dG(u, v) is the distance between u and v. In this paper, we first give the recurrence formulae for computing the hyper-Wiener indices of polyphenyl chains and polyphenyl spiders. We then obtain the sharp upper and lower bounds for the hyper-Wiener index among polyphenyl chains and polyphenyl spiders, respectively. Moreover, the corresponding extremal graphs are determined.

On the Largest Distance (Signless Laplacian) Eigenvalue of Non-transmission-regular Graphs

2018 ◽  
Vol 34 ◽  
pp. 459-471 ◽  
Author(s):  
Shuting Liu ◽  
Jinlong Shu ◽  
Jie Xue
Keyword(s):  
Spectral Radius ◽  
Regular Graph ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Regular Graphs ◽  
Laplacian Eigenvalue ◽  
Signless Laplacian ◽  
Sum Of Distances

Let $G=(V(G),E(G))$ be a $k$-connected graph with $n$ vertices and $m$ edges. Let $D(G)$ be the distance matrix of $G$. Suppose $\lambda_1(D)\geq \cdots \geq \lambda_n(D)$ are the $D$-eigenvalues of $G$. The transmission of $v_i \in V(G)$, denoted by $Tr_G(v_i)$ is defined to be the sum of distances from $v_i$ to all other vertices of $G$, i.e., the row sum $D_{i}(G)$ of $D(G)$ indexed by vertex $v_i$ and suppose that $D_1(G)\geq \cdots \geq D_n(G)$. The $Wiener~ index$ of $G$ denoted by $W(G)$ is given by $W(G)=\frac{1}{2}\sum_{i=1}^{n}D_i(G)$. Let $Tr(G)$ be the $n\times n$ diagonal matrix with its $(i,i)$-entry equal to $TrG(v_i)$. The distance signless Laplacian matrix of $G$ is defined as $D^Q(G)=Tr(G)+D(G)$ and its spectral radius is denoted by $\rho_1(D^Q(G))$ or $\rho_1$. A connected graph $G$ is said to be $t$-transmission-regular if $Tr_G(v_i) =t$ for every vertex $v_i\in V(G)$, otherwise, non-transmission-regular. In this paper, we respectively estimate $D_1(G)-\lambda_1(G)$ and $2D_1(G)-\rho_1(G)$ for a $k$-connected non-transmission-regular graph in different ways and compare these obtained results. And we conjecture that $D_1(G)-\lambda_1(G)>\frac{1}{n+1}$. Moreover, we show that the conjecture is valid for trees.

scholarly journals On the Wiener Index of the Dot Product Graph over Monogenic Semigroups

2020 ◽  
Vol 13 (5) ◽  
pp. 1231-1240
Author(s):  
Büşra Aydın ◽  
Nihat Akgüneş ◽  
İsmail Naci Cangül
Keyword(s):  
Graph Theory ◽  
Connected Graph ◽  
Molecular Graph ◽  
Wiener Index ◽  
Spectral Graph Theory ◽  
Topological Graph ◽  
Product Graph ◽  
Spectral Graph ◽  
Dot Product ◽  
Sum Of Distances

Algebraic study of graphs is a relatively recent subject which arose in two main streams: One is named as the spectral graph theory and the second one deals with graphs over several algebraic structures. Topological graph indices are widely-used tools in especially molecular graph theory and mathematical chemistry due to their time and money saving applications. The Wiener index is one of these indices which is equal to the sum of distances between all pairs of vertices in a connected graph. The graph over the nite dot product of monogenic semigroups has recently been dened and in this paper, some results on the Wiener index of the dot product graph over monogenic semigroups are given.

On the sum of the generalized distance eigenvalues of graphs

2020 ◽  
pp. 2050100
Author(s):  
Hilal A. Ganie ◽  
Abdollah Alhevaz ◽  
Maryam Baghipur
Keyword(s):  
Connected Graph ◽  
Independence Number ◽  
Distance Matrix ◽  
Wiener Index ◽  
Index Formula ◽  
Graph Invariants ◽  
Matrix Formula ◽  
Simple Connected Graph ◽  
Eigenvalues Of Graphs ◽  
Generalized Distance

In this paper, we study the generalized distance matrix [Formula: see text] assigned to simple connected graph [Formula: see text], which is the convex combinations of Tr[Formula: see text] and [Formula: see text] and defined as [Formula: see text] where [Formula: see text] and Tr[Formula: see text] denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph [Formula: see text], respectively. Denote with [Formula: see text], the generalized distance eigenvalues of [Formula: see text]. For [Formula: see text], let [Formula: see text] and [Formula: see text] be, respectively, the sum of [Formula: see text]-largest generalized distance eigenvalues and the sum of [Formula: see text]-smallest generalized distance eigenvalues of [Formula: see text]. We obtain bounds for [Formula: see text] and [Formula: see text] in terms of the order [Formula: see text], the Wiener index [Formula: see text] and parameter [Formula: see text]. For a graph [Formula: see text] of diameter 2, we establish a relationship between the [Formula: see text] and the sum of [Formula: see text]-largest generalized adjacency eigenvalues of the complement [Formula: see text]. We characterize the connected bipartite graph and the connected graphs with given independence number that attains the minimum value for [Formula: see text]. We also obtain some bounds for the graph invariants [Formula: see text] and [Formula: see text].

scholarly journals Ordering trees having small reverse wiener indices

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 637-648 ◽  
Author(s):  
Rundan Xing ◽  
Bo Zhou
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
The Third ◽  
Sum Of Distances

The Wiener index W(G) of a connected graph G is defined as the sum of distances between all unordered pairs of vertices of G. As a variation of the Wiener index, the reverse Wiener index of G is defined as ?(G) = ? n(n ? 1)d ? W(G), where n is the number of vertices, and d is the diameter of G. It is known that the star is the unique n-vertex tree with the smallest reverse Wiener index. We now determine the second and the third smallest reverse Wiener indices of n-vertex trees, and characterize the trees whose reverse Wiener indices attain these values for n ? 5.

scholarly journals On Generalized Distance Gaussian Estrada Index of Graphs

Symmetry ◽  
2019 ◽  
Vol 11 (10) ◽  
pp. 1276 ◽  
Author(s):  
Abdollah Alhevaz ◽  
Maryam Baghipur ◽  
Yilun Shang
Keyword(s):  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Quantum Information Theory ◽  
Extremal Graphs ◽  
Estrada Index ◽  
Graph Parameters ◽  
Generalized Distance

For a simple undirected connected graph G of order n, let D ( G ) , D L ( G ) , D Q ( G ) and T r ( G ) be, respectively, the distance matrix, the distance Laplacian matrix, the distance signless Laplacian matrix and the diagonal matrix of the vertex transmissions of G. The generalized distance matrix D α ( G ) is signified by D α ( G ) = α T r ( G ) + ( 1 - α ) D ( G ) , where α ∈ [ 0 , 1 ] . Here, we propose a new kind of Estrada index based on the Gaussianization of the generalized distance matrix of a graph. Let ∂ 1 , ∂ 2 , … , ∂ n be the generalized distance eigenvalues of a graph G. We define the generalized distance Gaussian Estrada index P α ( G ) , as P α ( G ) = ∑ i = 1 n e - ∂ i 2 . Since characterization of P α ( G ) is very appealing in quantum information theory, it is interesting to study the quantity P α ( G ) and explore some properties like the bounds, the dependence on the graph topology G and the dependence on the parameter α . In this paper, we establish some bounds for the generalized distance Gaussian Estrada index P α ( G ) of a connected graph G, involving the different graph parameters, including the order n, the Wiener index W ( G ) , the transmission degrees and the parameter α ∈ [ 0 , 1 ] , and characterize the extremal graphs attaining these bounds.

scholarly journals The reciprocal reverse Wiener index of unicyclic graphs

Filomat ◽  
2014 ◽  
Vol 28 (2) ◽  
pp. 249-255
Author(s):  
Jianping Li ◽  
Bo Zhou
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
Extremal Graphs ◽  
Unicyclic Graphs ◽  
Mathematical Chemistry

The reciprocal reverse Wiener index R?(G) of a connected graph G is defined in mathematical chemistry as the sum of weights 1/d(G)?dG(u,v) of all unordered pairs of distinct vertices u and v with dG(u, v) < d(G), where dG(u, v) is the distance between vertices u and v in G and d(G) is the diameter of G. We determine the minimum and maximum reciprocal reverse Wiener indices in the class of n-vertex unicyclic graphs and characterize the corresponding extremal graphs.

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