RECIPROCAL WIENER INDEX AND RECIPROCAL COMPLEMENTARY WIENER INDEX OF LINE GRAPHS (original) (raw)

On Wiener Index of Graphs and Their Line Graphs

2010

The Wiener index of a graph G, denoted by W (G), is the sum of distances between all pairs of vertices in G. In this paper, we consider the relation between the Wiener index of a graph, G, and its line graph, L(G). We show that if G is of minimum degree at least two, then W (G) ≤ W (L(G)). We prove that for every non-negative integer g 0 , there exists g > g 0 , such that there are infinitely many graphs G of girth g, satisfying W (G) = W (L(G)). This partially answers a question raised by Dobrynin and Mel'nikov [8] and encourages us to conjecture that the answer to a stronger form of their question is affirmative.

Some results on Wiener index of a graph: an overview

Proceedings of the 2nd Croatian Combinatorial Days, 2019

The Wiener index W (G) of a connected graph G is defined as the sum of distances between all pairs of vertices in G. In 1991,Šoltés [9] posed the problem of finding all graphs G such that equality W (G) = W (G − v) holds for all vertices v in G. The only known graph with this property is the cycle C 11. Our main object of study is the relaxed version of this problem: find graphs for which Wiener index does not change when a particular vertex v is removed. This overview contains results which were obtained and published during the past two years concerning relaxedŠoltés's problem.

An inequality between the edge-Wiener index and the Wiener index of a graph

Applied Mathematics and Computation, 2015

The Wiener index W (G) of a connected graph G is defined to be the sum u,v d(u, v) of distances between all unordered pairs of vertices in G. Similarly, the edge-Wiener index W e (G) of G is defined to be the sum e,f d(e, f) of distances between all unordered pairs of edges in G, or equivalently, the Wiener index of the line graph L(G). Wu [37] showed that W e (G) ≥ W (G) for graphs of minimum degree 2, where equality holds only when G is a cycle. Similarly, in [24] it was shown that W e (G) ≥ δ 2 −1 4 W (G) where δ denotes the minimum degree in G. In this paper, we extend/improve these two results by showing that W e (G) ≥ δ 2 4 W (G) with equality satisfied only if G is a path on 3 vertices or a cycle. Besides this, we also consider the upper bound for W e (G) as well as the ratio We(G) W (G). We show that among graphs G on n vertices We(G) W (G) attains its minimum for the star.

Terminal Wiener Index of Line Graphs

2013

The terminal Wiener index of a graph is defined as the sum of the distances between the pendent vertices of a graph. In this paper we obtain results for the terminal Wiener index of line graphs.

The Wiener index in iterated line graphs

Discrete Applied Mathematics, 2012

For a graph G, denote by L i (G) its i-iterated line graph and denote by W (G) its Wiener index. We prove that the function W (L i (G)) is convex in variable i. Moreover, this function is strictly convex if G is different from a path, a claw K 1,3 and a cycle. As an application we prove that W (L i (T)) = W (T) for every i ≥ 3 if T is a tree in which no leaf is adjacent to a vertex of degree 2, T = K 1 and T = K 2 .

Wiener Index of Total Graph of Some Graphs Research

2017

Let G = (V,E) be a graph. The total graph T (G) of G is that graph whose vertex set is V ∪ E, and two vertices are adjacent if and only if they are adjacent or incident in G. For a graph G = (V,E), the graph G.Sm is obtained by identifying each vertex of G by a root vertex of Sm and the graph Sm.G is obtained by identifying each vertex of Sm except root vertex by any vertex of G, where Sm is a star graph with m vertices. In this paper, we consider G as the cycle graph Cn with n vertices and investigate the Wiener index of the total graphs of Cn.Sm and Sn.Cm. MSC: 05C12, 05C76

Peripheral Wiener Index of a Graph

2017

The eccentricity of a vertex vvv is the maximum distance between vvv and anyother vertex. A vertex with maximum eccentricity is called a peripheral vertex.The peripheral Wiener index $ PW(G)$ of a graph GGG is defined as the sum ofthe distances between all pairs of peripheral vertices of G.G.G. In this paper, weinitiate the study of the peripheral Wiener index and we investigate its basicproperties. In particular, we determine the peripheral Wiener index of thecartesian product of two graphs and trees.

On a conjecture about the ratio of Wiener index in iterated line graphs

The Art of Discrete and Applied Mathematics, 2018

Let G be a graph. Denote by W (G) its Wiener index and denote by L i (G) its iiterated line graph. Dobrynin and Mel'nikov proposed to estimate the extremal values for the ratio R k (G) = W (L k (G))/W (G) for k ≥ 1. Motivated by this we study the ratio for higher k's. We prove that among all trees on n vertices the path P n has the smallest value of this ratio for k ≥ 3. We conjecture that this holds also for k = 2, and even more, for the class of all connected graphs on n vertices. Moreover, we conjecture that the maximum value of the ratio is obtained for the complete graph.