scholarly journals Wiener Index and Remoteness in Triangulations and Quadrangulations

2021 ◽  
Vol vol. 23 no. 1 (Graph Theory) ◽  
Author(s):  
Éva Czabarka ◽  
Peter Dankelmann ◽  
Trevor Olsen ◽  
László A. Székely
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
Upper Bounds ◽  
Arithmetic Mean ◽  
Asymptotic Formulae

Let $G$ be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If $\overline{\sigma}(v)$ denotes the arithmetic mean of the distances from $v$ to all other vertices of $G$, then the remoteness of $G$ is defined as the largest value of $\overline{\sigma}(v)$ over all vertices $v$ of $G$. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.

scholarly journals On the Sum of Distance Laplacian Eigenvalues of Graphs

2021 ◽  
Vol 54 ◽  
Author(s):  
Shariefuddin Pirzada ◽  
Saleem Khan
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Distance Matrix ◽  
Laplacian Matrix ◽  
Wiener Index ◽  
Upper Bounds ◽  
Diagonal Matrix ◽  
Laplacian Eigenvalues ◽  
Eigenvalues Of Graphs

Let $G$ be a connected graph with $n$ vertices, $m$ edges and having diameter $d$. The distance Laplacian matrix $D^{L}$ is defined as $D^L=$Diag$(Tr)-D$, where Diag$(Tr)$ is the diagonal matrix of vertex transmissions and $D$ is the distance matrix of $G$. The distance Laplacian eigenvalues of $G$ are the eigenvalues of $D^{L}$ and are denoted by $\delta_{1}, ~\delta_{1},~\dots,\delta_{n}$. In this paper, we obtain (a) the upper bounds for the sum of $k$ largest and (b) the lower bounds for the sum of $k$ smallest non-zero, distance Laplacian eigenvalues of $G$ in terms of order $n$, diameter $d$ and Wiener index $W$ of $G$. We characterize the extremal cases of these bounds. As a consequence, we also obtain the bounds for the sum of the powers of the distance Laplacian eigenvalues of $G$.

scholarly journals Two upper bounds for the degree distances of four sums of graphs

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 579-590 ◽  
Author(s):  
Mingqiang An ◽  
Liming Xiong ◽  
Kinkar Das
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Weighted Sum ◽  
Degree Of A Vertex ◽  
Degree Distance

The degree distance (DD), which is a weight version of the Wiener index, defined for a connected graph G as vertex-degree-weighted sum of the distances, that is, DD(G) = ?{u,v}?V(G)[dG(u)+dG(v)]d[u,v|G), where dG(u) denotes the degree of a vertex u in G and d(u,v|G) denotes the distance between two vertices u and v in G: In this paper, we establish two upper bounds for the degree distances of four sums of two graphs in terms of other indices of two individual graphs.

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 Upper bounds on the energy of graphs in terms of matching number

2021 ◽  
pp. 16-16
Author(s):  
Saieed Akbari ◽  
Abdullah Alazemi ◽  
Milica Andjelic
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Short Proof ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Maximum Matching ◽  
Matching Number ◽  
Open Problems

The energy of a graph G, ?(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. The matching number ?(G) is the number of edges in a maximum matching. In this paper, for a connected graph G of order n with largest vertex degree ? ? 6 we present two new upper bounds for the energy of a graph: ?(G) ? (n-1)?? and ?(G) ? 2?(G)??. The latter one improves recently obtained bound ?(G) ? {2?(G)?2?e + 1, if ?e is even; ?(G)(? a + 2?a + ?a-2?a), otherwise, where ?e stands for the largest edge degree and a = 2(?e + 1). We also present a short proof of this result and several open problems.

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.

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.

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.

scholarly journals Proper Vertex Connection and Graph Operations

2019 ◽  
Vol 19 (02) ◽  
pp. 1950001
Author(s):  
YINGYING ZHANG ◽  
XIAOYU ZHU
Keyword(s):  
Direct Product ◽  
Connected Graph ◽  
Upper Bounds ◽  
Lexicographic Product ◽  
Colored Graph ◽  
Connection Number ◽  
Graph Operations ◽  
Proper Vertex

A path in a vertex-colored graph is a vertex-proper path if any two internal adjacent vertices differ in color. A vertex-colored graph is proper vertex k-connected if any two vertices of the graph are connected by k disjoint vertex-proper paths of the graph. For a k-connected graph G, the proper vertex k-connection number of G, denoted by pvck(G), is defined as the smallest number of colors required to make G proper vertex k-connected. A vertex-colored graph is strong proper vertex-connected, if for any two vertices u, v of the graph, there exists a vertex-proper u-v geodesic. For a connected graph G, the strong proper vertex-connection number of G, denoted by spvc(G), is the smallest number of colors required to make G strong proper vertex-connected. In this paper, we study the proper vertex k-connection number and the strong proper vertex-connection number on the join of two graphs, the Cartesian, lexicographic, strong and direct product, and present exact values or upper bounds for these operations of graphs. Then we apply these results to some instances of Cartesian and lexicographic product networks.

scholarly journals The maximum Wiener index of maximal planar graphs

2020 ◽  
Vol 40 (4) ◽  
pp. 1121-1135
Author(s):  
Debarun Ghosh ◽  
Ervin Győri ◽  
Addisu Paulos ◽  
Nika Salia ◽  
Oscar Zamora
Keyword(s):  
Planar Graph ◽  
Connected Graph ◽  
Planar Graphs ◽  
Wiener Index ◽  
Maximal Planar Graph

Abstract 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 $$\lfloor \frac{1}{18}(n^3+3n^2)\rfloor $$ ⌊ 1 18 ( n 3 + 3 n 2 ) ⌋ . We prove this conjecture and determine the unique n-vertex maximal planar graph attaining this maximum, for every $$ n\ge 10$$ n ≥ 10 .

scholarly journals Some Bounds for the Kirchhoff Index of Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
Yujun Yang
Keyword(s):  
Connected Graph ◽  
Planar Graphs ◽  
Independence Number ◽  
Clique Number ◽  
Upper Bounds ◽  
Electrical Network ◽  
Lower And Upper Bounds ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Fullerene Graphs

The resistance distance between two vertices of a connected graphGis defined as the effective resistance between them in the corresponding electrical network constructed fromGby replacing each edge ofGwith a unit resistor. The Kirchhoff index ofGis the sum of resistance distances between all pairs of vertices. In this paper, general bounds for the Kirchhoff index are given via the independence number and the clique number, respectively. Moreover, lower and upper bounds for the Kirchhoff index of planar graphs and fullerene graphs are investigated.

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