scholarly journals Cycles, the Degree Distance, and the Wiener Index

2012 ◽  
Vol 02 (04) ◽  
pp. 156-159 ◽  
Author(s):  
Daniel Gray ◽  
Hua Wang
Keyword(s):  
Wiener Index ◽  
Degree Distance

Degree Distance of a Graph: A Degree Analog of the Wiener Index

1994 ◽  
Vol 34 (5) ◽  
pp. 1082-1086 ◽  
Author(s):  
Andrey A. Dobrynin ◽  
Amide A. Kochetova
Keyword(s):  
Wiener Index ◽  
Degree Distance

The Wiener index, the hyper-Wiener index and the degree distance index of the corona Cm.Cn

2014 ◽  
Vol 8 ◽  
pp. 4217-4226
Author(s):  
Mohamed Essalih ◽  
Mohamed El Marraki ◽  
Abd Errahmane Atmani
Keyword(s):  
Wiener Index ◽  
Degree Distance

scholarly journals Some distance based indices of graphs based on four new operations related to the lexicographic product

2019 ◽  
Vol 11 (2) ◽  
pp. 258-267
Author(s):  
N. Dehgardi ◽  
S.M. Sheikholeslami ◽  
M. Soroudi
Keyword(s):  
Molecular Graph ◽  
Wiener Index ◽  
Lexicographic Product ◽  
Total Graph ◽  
Degree Of A Vertex ◽  
Degree Distance

For a (molecular) graph, the Wiener index, hyper-Wiener index and degree distance index are defined as $$W(G)= \sum_{\{u,v\}\subseteq V(G)}d_G(u,v),$$ $$WW(G)=W(G)+\sum_{\{u,v\}\subseteq V(G)} d_{G}(u,v)^2,$$ and $$DD(G)=\sum_{\{u,v\}\subseteq V(G)}d_G(u, v)(d(u/G)+d(v/G)),$$ respectively, where $d(u/G)$ denotes the degree of a vertex $u$ in $G$ and $d_G(u, v)$ is distance between two vertices $u$ and $v$ of a graph $G$. In this paper, we study Wiener index, hyper-Wiener index and degree distance index of graphs based on four new operations related to the lexicographic product, subdivision and total graph.

Distance-based indices of complete m-ary trees

2020 ◽  
Vol 12 (04) ◽  
pp. 2050041
Author(s):  
Mesfin Masre ◽  
Samuel Asefa Fufa ◽  
Tomáš Vetrík
Keyword(s):  
Computer Science ◽  
Recurrence Relations ◽  
Wiener Index ◽  
The Other ◽  
Binary Trees ◽  
New Methods ◽  
Degree Distance ◽  
Gutman Index ◽  
Eccentric Distance

Binary and [Formula: see text]-ary trees have extensive applications, particularly in computer science and chemistry. We present exact values of all important distance-based indices for complete [Formula: see text]-ary trees. We solve recurrence relations to obtain the value of the most well-known index called the Wiener index. New methods are used to express the other indices (the degree distance, the eccentric distance sum, the Gutman index, the edge-Wiener index, the hyper-Wiener index and the edge-hyper-Wiener index) as well. Values of distance-based indices for complete binary trees are corollaries of the main results.

scholarly journals WIENER INDEX OF TREES OF GIVEN ORDER AND DIAMETER AT MOST

2013 ◽  
Vol 89 (3) ◽  
pp. 379-396 ◽  
Author(s):  
SIMON MUKWEMBI ◽  
TOMÁŠ VETRÍK
Keyword(s):  
Upper Bound ◽  
Open Problem ◽  
Wiener Index ◽  
Upper Bounds ◽  
Degree Distance ◽  
Gutman Index

AbstractThe long-standing open problem of finding an upper bound for the Wiener index of a graph in terms of its order and diameter is addressed. Sharp upper bounds are presented for the Wiener index, and the related degree distance and Gutman index, for trees of order$n$and diameter at most$6$.

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 The Sum Degree Distance and the Product Degree Distance of Generalized Transformation Graphs Gab

2016 ◽  
Vol 16 ◽  
pp. 76-88
Author(s):  
Keerthi G. Mirajkar ◽  
Y.B. Priyanka
Keyword(s):  
Wiener Index ◽  
Weighted Version ◽  
Line Splitting ◽  
Degree Distance ◽  
Splitting Graph

In this contribution, we consider line splitting graph LS(G) of a graph G as transformation graph G++ of Gab. We investigate the sum degree distance DD+(G) and product degree distance DD*(G) of transformation graph Gab, which are weighted version of Wiener index. The Transformation graphs of Gab are G++, G+-, G-+ and G--.

scholarly journals Computing Zagreb, Hyper-Wiener and Degree-Distance indices of four new sums of graphs

2011 ◽  
Vol 27 (2) ◽  
pp. 153-164
Author(s):  
A. R. ASHRAFI ◽  
◽  
A. HAMZEH ◽  
S. HOSSEIN-ZADEH ◽  
◽  
...  
Keyword(s):  
Wiener Index ◽  
Graph Operations ◽  
Degree Distance ◽  
Appl Math

Eliasi and Taeri [M. Eliasi and B. Taeri, Four new sums of graphs and their Wiener indices, Discrete Appl. Math., 157 (2009), 794-803] presented four new sums and computed their Wiener index. In this paper, we continue this work to compute the Zagreb, Hyper-Wiener and Degree-Distance Indices of these graph operations. Some applications are also presented.

Further results on Zagreb eccentricity coindices

2020 ◽  
Vol 12 (06) ◽  
pp. 2050075
Author(s):  
Mahdieh Azari
Keyword(s):  
Wiener Index ◽  
Connectivity Index ◽  
Lower And Upper Bounds ◽  
Graph Invariants ◽  
Eccentric Connectivity Index ◽  
Size Number ◽  
Connectivity Indices ◽  
Degree Distance ◽  
Graph Parameters ◽  
Eccentric Distance

The eccentric connectivity index and second Zagreb eccentricity index are well-known graph invariants defined as the sums of contributions dependent on the eccentricities of adjacent vertices over all edges of a connected graph. The coindices of these invariants have recently been proposed by considering analogous contributions from the pairs of non-adjacent vertices. Here, we obtain several lower and upper bounds on the eccentric connectivity coindex and second Zagreb eccentricity coindex in terms of some graph parameters such as order, size, number of non-adjacent vertex pairs, radius, and diameter, and relate these invariants to some well-known graph invariants such as Zagreb indices and coindices, status connectivity indices and coindices, ordinary and multiplicative Zagreb eccentricity indices, Wiener index, degree distance, total eccentricity, eccentric connectivity index, second eccentric connectivity index, and eccentric-distance sum. Moreover, we compute the values of these coindices for two graph constructions, namely, double graphs and extended double graphs.

scholarly journals Remarks on Distance Based Topological Indices for ℓ-Apex Trees

Symmetry ◽  
2020 ◽  
Vol 12 (5) ◽  
pp. 802
Author(s):  
Martin Knor ◽  
Muhammad Imran ◽  
Muhammad Kamran Jamil ◽  
Riste Škrekovski
Keyword(s):  
Graph Theory ◽  
Wiener Index ◽  
Extreme Value ◽  
Topological Indices ◽  
The Other ◽  
Harary Index ◽  
Open Questions ◽  
Degree Distance ◽  
The One ◽  
Extremal Values

A graph G is called an ℓ-apex tree if there exist a vertex subset A ⊂ V ( G ) with cardinality ℓ such that G − A is a tree and there is no other subset of smaller cardinality with this property. In the paper, we investigate extremal values of several monotonic distance-based topological indices for this class of graphs, namely generalized Wiener index, and consequently for the Wiener index and the Harary index, and also for some newer indices as connective eccentricity index, generalized degree distance, and others. For the one extreme value we obtain that the extremal graph is a join of a tree and a clique. Regarding the other extreme value, which turns out to be a harder problem, we obtain results for ℓ = 1 and pose some open questions for higher ℓ. Symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including topological indices of graphs.

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