scholarly journals On the reformulated reciprocal degree distance of graphs

2016 ◽  
Vol 25 (2) ◽  
pp. 205-213
Author(s):  
K. PATTABIRAMAN ◽  
◽  
M. VIJAYARAGAVAN ◽  
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Vertex Degree ◽  
Symmetric Difference ◽  
Weighted Sum ◽  
Graph Invariant ◽  
Harary Index ◽  
Degree Distance

The reciprocal degree distance (RDD), defined for a connected graph G as vertex-degree-weighted sum of the reciprocal distances, that is, RDD(G) = P u,v∈V (G) (d(u)+d(v)) dG(u,v) . The new graph invariant named reformulated reciprocal degree distance is defined for a connected graph G as Rt(G) = P u,v∈V (G) (d(u)+d(v)) dG(u,v)+t , t ≥ 0. The reformulated reciprocal degree distance is a weight version of the t-Harary index, that is, Ht(G) = P u,v∈V (G) 1 dG(u,v)+t , t ≥ 0. In this paper, the reformulated reciprocal degree distance and reciprocal degree distance of disjunction, symmetric difference, Cartesian product of two graphs are obtained. Finally, we obtain the reformulated reciprocal degree distance and reciprocal degree distance of double a graph.

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 Reciprocal product-degree distance of graphs

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2217-2231
Author(s):  
Guifu Su ◽  
Liming Xiong ◽  
Ivan Gutman ◽  
Lan Xu
Keyword(s):  
Connected Graph ◽  
Upper Bounds ◽  
Graph Invariant ◽  
Lower And Upper Bounds ◽  
Harary Index ◽  
Underlying Graph ◽  
Degree Distance ◽  
Degree Modification ◽  
Type Relation

We investigate a new graph invariant named reciprocal product-degree distance, defined as: RDD* = ?{u,v}?V(G)u?v deg(u)?deg(v)/dist(u,v) where deg(v) is the degree of the vertex v, and dist(u,v) is the distance between the vertices u and v in the underlying graph. RDD* is a product-degree modification of the Harary index. We determine the connected graph of given order with maximum RDD*-value, and establish lower and upper bounds for RDD*. Also a Nordhaus-Gaddum-type relation for RDD* is obtained.

scholarly journals Reformulated Reciprocal Degree Distance and Reciprocal Degree Distance of the Complement of the Mycielskian Graph and Generalized Mycielskian

2019 ◽  
Vol 2019 ◽  
pp. 1-8
Author(s):  
Feifei Zhao ◽  
Hong Bian ◽  
Haizheng Yu ◽  
Min Liu
Keyword(s):  
Connected Graph ◽  
Harary Index ◽  
Degree Distance

The reformulated reciprocal degree distance is defined for a connected graph G as R¯t(G)=(1/2)∑u,υ∈VG((dG(u)+dG(υ))/(dG(u,υ)+t)),t≥0, which can be viewed as a weight version of the t-Harary index; that is, H¯t(G)=(1/2)∑u,υ∈VG(1/(dG(u,υ)+t)),t≥0. In this paper, we present the reciprocal degree distance index of the complement of Mycielskian graph and generalize the corresponding results to the generalized Mycielskian graph.

scholarly journals The Reciprocal Complementary Wiener Number of Graph Operations

2021 ◽  
Vol 45 (01) ◽  
pp. 139-154
Author(s):  
R. NASIRI ◽  
A. NAKHAEI ◽  
A. R. SHOJAEIFARD
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Symmetric Difference ◽  
Graph Operations ◽  
Strong Product ◽  
Product Composition ◽  
Wiener Number ◽  
Corona Product

The reciprocal complementary Wiener number of a connected graph G is defined as ∑ {x,y}⊆V (G) 1 D+1-−-dG(x,y), where D is the diameter of G and dG(x,y) is the distance between vertices x and y. In this work, we study the reciprocal complementary Wiener number of various graph operations such as join, Cartesian product, composition, strong product, disjunction, symmetric difference, corona product, splice and link of graphs.

scholarly journals Computing Exact Values for Gutman Indices of Sum Graphs under Cartesian Product

2021 ◽  
Vol 2021 ◽  
pp. 1-20
Author(s):  
Abdulaziz Mohammed Alanazi ◽  
Faiz Farid ◽  
Muhammad Javaid ◽  
Augustine Munagi
Keyword(s):  
Connected Graph ◽  
Topological Index ◽  
Cartesian Product ◽  
Chemical Compounds ◽  
Factor Graphs ◽  
Cartesian Products ◽  
Degree Distance ◽  
Gutman Index ◽  
Extremal Theory ◽  
Theory Of Graphs

Gutman index of a connected graph is a degree-distance-based topological index. In extremal theory of graphs, there is great interest in computing such indices because of their importance in correlating the properties of several chemical compounds. In this paper, we compute the exact formulae of the Gutman indices for the four sum graphs (S-sum, R-sum, Q-sum, and T-sum) in the terms of various indices of their factor graphs, where sum graphs are obtained under the subdivision operations and Cartesian products of graphs. We also provide specific examples of our results and draw a comparison with previously known bounds for the four sum graphs.

scholarly journals Steiner Degree Distance of Two Graph Products

2019 ◽  
Vol 27 (2) ◽  
pp. 83-99 ◽  
Author(s):  
Yaping Mao ◽  
Zhao Wang ◽  
Kinkar Ch. Das
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Graph Products ◽  
Product Graphs ◽  
Degree Distance ◽  
Cartesian Product Graphs

AbstractThe degree distance DD(G) of a connected graph G was invented by Dobrynin and Kochetova in 1994. Recently, one of the present authors introduced the concept of k-center Steiner degree distance defined as SDD_k (G) = \sum\limits_{\mathop {S \subseteq V(G)}\limits_{\left| S \right| = k} } {\left[ {\sum\limits_{v \in S} {{\it deg} _G (v)} } \right]d_G (S),} where dG(S) is the Steiner k-distance of S and degG(v) is the degree of the vertex v in G. In this paper, we investigate the Steiner degree distance of complete and Cartesian product graphs.

scholarly journals THE GENERAL POSITION NUMBER OF THE CARTESIAN PRODUCT OF TWO TREES

2020 ◽  
pp. 1-10
Author(s):  
JING TIAN ◽  
KEXIANG XU ◽  
SANDI KLAVŽAR
Keyword(s):  
Shortest Path ◽  
Connected Graph ◽  
General Position ◽  
Cartesian Product

Abstract The general position number of a connected graph is the cardinality of a largest set of vertices such that no three pairwise-distinct vertices from the set lie on a common shortest path. In this paper it is proved that the general position number is additive on the Cartesian product of two trees.

ON MOSTAR INDEX OF GRAPHS

2021 ◽  
Vol 10 (4) ◽  
pp. 2115-2129
Author(s):  
P. Kandan ◽  
S. Subramanian
Keyword(s):  
Connected Graph ◽  
Cartesian Product ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Great Success ◽  
Complete Graphs ◽  
Molecular Graphs ◽  
Graphs And Networks ◽  
Complete Bipartite Graphs ◽  
Simple Connected Graph

On the great success of bond-additive topological indices like Szeged, Padmakar-Ivan, Zagreb, and irregularity measures, yet another index, the Mostar index, has been introduced recently as a peripherality measure in molecular graphs and networks. For a connected graph G, the Mostar index is defined as $$M_{o}(G)=\displaystyle{\sum\limits_{e=gh\epsilon E(G)}}C(gh),$$ where $C(gh) \,=\,\left|n_{g}(e)-n_{h}(e)\right|$ be the contribution of edge $uv$ and $n_{g}(e)$ denotes the number of vertices of $G$ lying closer to vertex $g$ than to vertex $h$ ($n_{h}(e)$ define similarly). In this paper, we prove a general form of the results obtained by $Do\check{s}li\acute{c}$ et al.\cite{18} for compute the Mostar index to the Cartesian product of two simple connected graph. Using this result, we have derived the Cartesian product of paths, cycles, complete bipartite graphs, complete graphs and to some molecular graphs.

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 Extremal Bipartite Graphs with Given Parameters on the Resistance–Harary Index

Symmetry ◽  
2019 ◽  
Vol 11 (5) ◽  
pp. 615
Author(s):  
Hongzhuan Wang ◽  
Piaoyang Yin
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Extremal Graphs ◽  
Graph Invariant ◽  
Resistance Distance ◽  
Harary Index ◽  
H Index ◽  
Electronic Networks ◽  
Minimum Number ◽  
Graph Parameters

Resistance distance is a concept developed from electronic networks. The calculation of resistance distance in various circuits has attracted the attention of many engineers. This report considers the resistance-based graph invariant, the Resistance–Harary index, which represents the sum of the reciprocal resistances of any vertex pair in the figure G, denoted by R H ( G ) . Vertex bipartiteness in a graph G is the minimum number of vertices removed that makes the graph G become a bipartite graph. In this study, we give the upper bound and lower bound of the R H index, and describe the corresponding extremal graphs in the bipartite graph of a given order. We also describe the graphs with maximum R H index in terms of graph parameters such as vertex bipartiteness, cut edges, and matching numbers.

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