Reciprocal Degree Distance Research Articles

Sufficient conditions for [formula omitted]-leaf-connected graphs in terms of the first Zagreb index, the reciprocal degree distance and the forgotten topological index

For integer k≥2, a graph G is called k-leaf-connected if |V(G)|≥k+1 and given any subset S⊆V(G) with |S|=k, G always has a spanning tree T such that S is precisely the set of leaves of T. Thus a graph is 2-leaf-connected if and only if it is Hamiltonian-connected. The first Zagreb index M1(G) is equal to the sum of squares of the degrees of the vertices of the underlying molecular graph G. The reciprocal degree distance (RDD), defined as vertex-degree-weighted sum of the reciprocal distances, that is, RDD(G)=∑vi≠vjdegG(vi)+degG(vj)dG(vi,vj), where degG(vi) is the degree of the vertex vi in G and dG(vi,vj) denotes the distance between two vertices vi and vj in G. The forgotten topological index of G is the sum of cubes of all its vertex degrees, which plays a significant role in measuring the branching of the carbon atom skeleton. Finding sufficient conditions for graphs possessing certain properties is an important and meaningful problem. In this paper, we give sufficient conditions in terms of the first Zagreb index, the reciprocal degree distance or the forgotten topological index for a graph to be k-leaf-connected, which generalizes two results of An (2022). All of these results are best possible.

Reciprocal degree distance and Hamiltonian properties of graphs

For a connected graph G, the reciprocal degree distance (RDD) of the graph G is defined as; RDD(G)=∑u≠vdegG⁡(u)+degG⁡(v)dG(u,v), where degG⁡(u) is the degree of u in G and dG(u,v) is the distance between u and v in G. In this article, we give sufficient condition for a graph to be traceable, Hamiltonian, Hamiltonian-connected in terms of RDD(G).

Steiner Distance Related Parameters of Wheel graphs

For a connected graph of order at least and the Steiner distance among the vertices of is the minimum size among all the connected subgraphs whose vertex sets contain In this paper, we calculate the Steiner distance parameters such as Steiner degree distance, Steiner Gutman index, Steiner Harary index and Steiner reciprocal degree distance of a wheel graphs.

On reciprocal degree distance of graphs

Given a connected graph H, its reciprocal degree distance is defined asRDD(H)=∑x≠ydH(vx)+dH(vy)dH(vx,vy), where dH(vx) denotes the degree of the vertex vx in the graph H and dH(vx,vy) is the shortest distance between vx and vy in H. The goal of this paper is to establish some sufficient conditions to judge that a graph to be ħ-hamiltonian, ħ-path-coverable or ħ-edge-hamiltonian by employing the reciprocal degree distance.

Reciprocal Degree Distance of Eliasi-Taeri Sums of Graphs

Reciprocal Degree Distance of Eliasi-Taeri Sums of Graphs

The first Zagreb index, reciprocal degree distance and Hamiltonian-connectedness of graphs

Let G be a connected graph. The first Zagreb index M1(G) is equal to the sum of squares of the degrees of the vertices of G. The reciprocal degree distance (RDD), defined as vertex-degree-weighted sum of the reciprocal distances, that is, RDD(G)=∑vi≠vjdegG(vi)+degG(vj)dG(vi,vj), where degG(vi) is the degree of the vertex vi in G and dG(vi,vj) denotes the distance between two vertices vi and vj in G. Finding sufficient conditions for graphs possessing certain properties is an important and meaningful problem. In this paper, we give sufficient conditions in terms of the first Zagreb index or the reciprocal degree distance for a graph to be Hamiltonian-connected.

Degree distance based topological indices of some graph transforms

A topological index is invariantly used to study the structure of molecular graph. In this paper, we have generalized degree distance index, Gutman index and reciprocal degree distance index, and calculate generalized indices for splitting and co-splitting graphs.

Some More Results on Reciprocal Degree Distance Index and ℱ‐Sum Graphs

A chemical invariant of graphical structure is a unique value characteristic that remains unchanged under graph automorphisms. In the study of QSAR/QSPR, like many other chemical invariants, reciprocal degree distance has played a significant role to estimate the bioactivity of several compounds in chemistry. Reciprocal degree distance is a chemical invariant, which is the degree weighted version of Harary index, i.e., . Eliasi and Taeri proposed four new graphic unary operations: , and , frequently implemented in sum of graphs, symbolized as , i.e., sum of two graphs , is one of the unary graphic operations . This work provides constraints for the above‐mentioned invariant for this binary graphic operation F‐sum of graphs.

Inverse and Hyper Zagreb Indices: Application to Quantitative Structure Properties Relationship

The topology index can be derived from molecular connectivity, which can be related to the chemical or physical properties of molecules. The topology indices of alkanes are calculated using the modified-Zagreb index, direct and reciprocal degree distance indices, and the modified-Zagreb index. These indices are tested as the structural descriptor for thermodynamic properties such as boiling point (bp), heat vaporization at 25oC (hv), critical temperature (ct), critical pressure (PC) and magnetooptical properties. Multiple linear regression analyses were done to find the best fitting between these indices and the thermodynamic properties. The results indicate that the boiling point, critical temperature, and heat vapor are reasonably well accounted for by the presence of more than two parameters. The regression equation for the four thermodynamic properties and magnetooptical properties shows a good relation property.

A note on Steiner reciprocal degree distance

The concept of reciprocal degree distance [Formula: see text] of a connected graph [Formula: see text] was introduced in 2012. The Steiner distance in a graph, introduced by Chartrand et al. in 1989, is a natural generalization of the concept of classical graph distance. The [Formula: see text]-center Steiner reciprocal degree distance defined as [Formula: see text], where [Formula: see text] is the Steiner [Formula: see text]-distance of [Formula: see text] and [Formula: see text] is the degree of the vertex [Formula: see text] in [Formula: see text]. Motivated from Zhang’s paper [X. Zhang, Reciprocal Steiner degree distance, Utilitas Math., accepted for publication], we find the expression for [Formula: see text] of complete bipartite graphs. Also, we give a straightforward method to compute Steiner Gutman index and Steiner degree distance of path.

Comparative study of distance-based graph invariants

The investigation on relationships between various graph invariants has received much attention over the past few decades, and some of these research are associated with Graffiti conjectures (Fajtlowicz and Waller in Congr Numer 60:187–197, 1987) or AutoGraphiX conjectures (Aouchiche et al. in: Liberti, Maculan (eds) Global optimization: from theory to implementation, Springer, New York, 2006). The reciprocal degree distance (RDD), the adjacent eccentric distance sum (AEDS), the average distance (AD) and the connective eccentricity index (CEI) are all distance-based graph invariants or topological indices, some of which found applications in Chemistry. In this paper, we investigate the relationship between RDD and other three graph invariants AEDS, CEI and AD. First, we prove that AEDS > RDD for any tree with at least three vertices. Then, we prove that RDD > CEI for all connected graphs with at least three vertices. Moreover, we prove that RDD > AD for all connected graphs with at least three vertices. As a consequence, we prove that AEDS > CEI and AEDS > AD for any tree with at least three vertices.

Distance based topological descriptors for two classes of graphs

In this paper, the exact formula for the generalized product degree distance, reciprocal product degree distance and product degree distance of Mycielskian graph and its complement are obtained. In addition, we compute the above indices for non-commuting graph.

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

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.

Reciprocal degree distance and graph properties

The reciprocal degree distance (RDD), defined for a connected graph G as vertex-degree- weighted sum of the reciprocal distances, that is, RDD(G)=∑u≠vdG(u)+dG(v)dG(u,v), where dG(u) is the degree of the vertex u in the graph G and dG(u,v) denotes the distance between two vertices u and v in the graph G. The reciprocal degree distance is a weight version of the Harary index, just as the degree distance is a weight version of the Wiener index. Finding sufficient conditions for graphs possessing certain properties is an important and meaningful problem. In this paper, we give sufficient conditions for a graph to be k-connected or β-deficient in terms of the reciprocal degree distance.

Reformulated Reciprocal Degree Distance of Transformation Graph

The reformulated reciprocal degree distance is defined for a connected graph G as Rt‾(G)=12∑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)=12∑u,v∈V(G)1dG(u,v)+t,t≥0. In this paper, the reformulated reciprocal degree distance and reciprocal degree distance of the Mycielskian graph and the complement of the Mycielskian graph are obtained.

On the reformulated reciprocal degree distance of graphs

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.

On Topological Indices of Graph Transformation

In this paper, the exact formulae for the generalized degree distance, reciprocal degree distance and degree distance of Mycielskian graph and the complement of the Mycielskian graph are obtained. Also, we obtain the sharp bounds for the ABC index, GA index, third Zagreb index and third Zagreb coindex of the Mycielskian graph.

Reformulated Reciprocal Degree Distance of Graph Operations

The reciprocal degree distance (RDD), defined for a connected graph G as vertex-degree-weighted sum of the reciprocal distances, that is, \(RDD(G) =\sum \nolimits _{u,v\in V(G)}\frac{(d(u) + d(v))}{d_G(u,v)}.\) The new graph invariant named reformulated reciprocal degree distance is defined for a connected graph G as \(\overline{R}_t(G) =\sum \nolimits _{u,v\in V(G)}\frac{(d(u) + d(v))}{d_G(u,v)+t},~t\ge 0.\) The reformulated reciprocal degree distance is a weight version of the t-Harary index, that is, \(\overline{H}_t(G) =\sum \nolimits _{u,v\in V(G)}\frac{1}{d_G(u,v)+t},~t\ge 0.\) In this paper, the reformulated reciprocal degree distance and reciprocal degree distance of composition, join, tensor product and strong product of two graphs are obtained.

Generalized Degree Distance of Strong Product of Graphs

In this paper, the exact formulae for the generalized degree distance, degree distance and reciprocal degree distance of strong product of a connected and the complete multipartite graph with partite sets of sizes m0, m1, ..., mr−1 are obtained. Using the results obtained here, the formulae for the degree distance and reciprocal degree distance of the closed and open fence graphs are computed.

Reciprocal degree distance of product graphs

The reciprocal degree distance (RDD), defined for a connected graph G as vertex-degree-weighted sum of the reciprocal distances, that is, RDD(G)=∑u,v∈V(G)(d(u)+d(v))dG(u,v). The reciprocal degree distance is a weight version of the Harary index, just as the degree distance is a weight version of the Wiener index. In this paper, the exact formulae for the reciprocal degree distance of tensor product G×Km0,m1,…,mr−1 and the strong product G⊠Km0,m1,…,mr−1, where Km0,m1,…,mr−1 is the complete multipartite graph with partite sets of sizes m0,m1,…,mr−1 are obtained. Finally, we apply our result to compute the reciprocal degree distance of open fence and closed fence graphs.