scholarly journals On the Kirchhoff and the Wiener Indices of Graphs and Block Decomposition

10.37236/3508 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Ashkan Nikseresht ◽  
Zahra Sepasdar
Keyword(s):  
Connected Graph ◽  
Rooted Tree ◽  
Wiener Index ◽  
Block Decomposition ◽  
Kirchhoff Index ◽  
Simple Connected Graph

In this article we state a relation between the Kirchhoff and Wiener indices of a simple connected graph $G$ and the Kirchhoff and Wiener indices of those subgraphs of $G$ which are induced by its blocks. Then as an application, we define a composition of a rooted tree $T$ and a graph $G$ and calculate its Kirchhoff index in terms of parameters of $T$ and $G$. Finally, we present an algorithm for computing the resistance distances and the Kirchhoff index and a similar one for computing the weighted distances and the Wiener index of a graph. These algorithms are asymptotically faster than the previously known algorithms, on graphs in which the order of the subgraphs induced by blocks is small with respect to the order of the graph.

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.

SOME NEW LOWER BOUNDS FOR THE KIRCHHOFF INDEX OF A GRAPH

2017 ◽  
Vol 97 (1) ◽  
pp. 1-10
Author(s):  
I. MILOVANOVIĆ ◽  
M. MATEJIĆ ◽  
E. GLOGIĆ ◽  
E. MILOVANOVIĆ
Keyword(s):  
Lower Bounds ◽  
Connected Graph ◽  
Topological Index ◽  
Its Sequence ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Vertex Degrees ◽  
Simple Connected Graph

Let$G$be a simple connected graph with$n$vertices and$m$edges and$d_{1}\geq d_{2}\geq \cdots \geq d_{n}>0$its sequence of vertex degrees. If$\unicode[STIX]{x1D707}_{1}\geq \unicode[STIX]{x1D707}_{2}\geq \cdots \geq \unicode[STIX]{x1D707}_{n-1}>\unicode[STIX]{x1D707}_{n}=0$are the Laplacian eigenvalues of$G$, then the Kirchhoff index of$G$is$\mathit{Kf}(G)=n\sum _{i=1}^{n-1}\unicode[STIX]{x1D707}_{i}^{-1}$. We prove some new lower bounds for$\mathit{Kf}(G)$in terms of some of the parameters$\unicode[STIX]{x1D6E5}=d_{1}$,$\unicode[STIX]{x1D6E5}_{2}=d_{2}$,$\unicode[STIX]{x1D6E5}_{3}=d_{3}$,$\unicode[STIX]{x1D6FF}=d_{n}$,$\unicode[STIX]{x1D6FF}_{2}=d_{n-1}$and the topological index$\mathit{NK}=\prod _{i=1}^{n}d_{i}$.

On the sum of the generalized distance eigenvalues of graphs

2020 ◽  
pp. 2050100
Author(s):  
Hilal A. Ganie ◽  
Abdollah Alhevaz ◽  
Maryam Baghipur
Keyword(s):  
Connected Graph ◽  
Independence Number ◽  
Distance Matrix ◽  
Wiener Index ◽  
Index Formula ◽  
Graph Invariants ◽  
Matrix Formula ◽  
Simple Connected Graph ◽  
Eigenvalues Of Graphs ◽  
Generalized Distance

In this paper, we study the generalized distance matrix [Formula: see text] assigned to simple connected graph [Formula: see text], which is the convex combinations of Tr[Formula: see text] and [Formula: see text] and defined as [Formula: see text] where [Formula: see text] and Tr[Formula: see text] denote the distance matrix and diagonal matrix of the vertex transmissions of a simple connected graph [Formula: see text], respectively. Denote with [Formula: see text], the generalized distance eigenvalues of [Formula: see text]. For [Formula: see text], let [Formula: see text] and [Formula: see text] be, respectively, the sum of [Formula: see text]-largest generalized distance eigenvalues and the sum of [Formula: see text]-smallest generalized distance eigenvalues of [Formula: see text]. We obtain bounds for [Formula: see text] and [Formula: see text] in terms of the order [Formula: see text], the Wiener index [Formula: see text] and parameter [Formula: see text]. For a graph [Formula: see text] of diameter 2, we establish a relationship between the [Formula: see text] and the sum of [Formula: see text]-largest generalized adjacency eigenvalues of the complement [Formula: see text]. We characterize the connected bipartite graph and the connected graphs with given independence number that attains the minimum value for [Formula: see text]. We also obtain some bounds for the graph invariants [Formula: see text] and [Formula: see text].

scholarly journals Comparison between Laplacian--energy--like invariant and Kirchhoff index

2016 ◽  
Vol 31 ◽  
pp. 27-41 ◽  
Author(s):  
Shariefuddin Pirzada ◽  
Hilal Ganie ◽  
Ivan Gutman
Keyword(s):  
Connected Graph ◽  
Sufficient Conditions ◽  
Kirchhoff Index ◽  
Laplacian Eigenvalues ◽  
Laplacian Energy ◽  
Simple Connected Graph

For a simple connected graph G of order n, having Laplacian eigenvalues μ_1, μ_2, . . . ,μ_{n−1}, μ_n = 0, the Laplacian–energy–like invariant (LEL) and the Kirchhoff index (Kf) are defined as LEL(G) = \sum_{i=1}^{n-1} \sqrt{μ_i} Kf(G) = \sum_{i=1}^{n-1} 1/μ_i, respectively. In this paper, LEL and Kf arecompared, and sufficient conditions for the inequality Kf(G) < LEL(G) are established.

Kirchhoff Index of Cyclopolyacenes

2010 ◽  
Vol 65 (10) ◽  
pp. 865-870 ◽  
Author(s):  
Yan Wang ◽  
Wenwen Zhang
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
Effective Resistance ◽  
Laplacian Spectrum ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Hexagonal Chain

The resistance distance between two vertices of a connected graph G is computed as the effective resistance between them in the corresponding network constructed from G by replacing each edge with a unit resistor. The Kirchhoff index of G is the sum of resistance distances between all pairs of vertices. In this paper, following the method of Y. J. Yang and H. P. Zhang in the proof of the Kirchhoff index of the linear hexagonal chain, we obtain the Kirchhoff index of cyclopolyacenes, denoted by HRn, in terms of its Laplacian spectrum. We show that the Kirchhoff index of HRnis approximately one third of its Wiener index.

scholarly journals On the Edge Wiener index

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 541-549
Author(s):  
Abolghasem Soltani ◽  
Ali Iranmanesh
Keyword(s):  
Explicit Formula ◽  
Connected Graph ◽  
Line Graph ◽  
Cartesian Product ◽  
Wiener Index ◽  
New Method ◽  
Automorphisms Of Graphs ◽  
Group Automorphisms ◽  
Sum Of Distances ◽  
Simple Connected Graph

Let G be a simple connected graph. The Wiener index of G is the sum of all distances between vertices of G. Whereas, the edge Wiener index of G is defined as the sum of distances between all pairs of edges of G where the distance between the edges f and g in E(G) is defined as the distance between the vertices f and g in the line graph of G. In this paper we will describe a new method for calculating the edge Wiener index. Then find this index for the triangular graphs. Also, we obtain an explicit formula for the Wiener index of the Cartesian product of two graphs using the group automorphisms of graphs.

scholarly journals On the edge-Wiener index of the disjunctive product of simple graphs

2020 ◽  
Vol 30 (1) ◽  
pp. 1-14
Author(s):  
M. Azari ◽  
◽  
A. Iranmanesh ◽  
Keyword(s):  
Connected Graph ◽  
Line Graph ◽  
Wiener Index ◽  
Simple Graphs ◽  
Sum Of Distances ◽  
Simple Connected Graph ◽  
Product Of Graphs

The edge-Wiener index of a simple connected graph G is defined as the sum of distances between all pairs of edges of G where the distance between two edges in G is the distance between the corresponding vertices in the line graph of G. In this paper, we study the edge-Wiener index under the disjunctive product of graphs and apply our results to compute the edge-Wiener index for the disjunctive product of paths and cycles.

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.

Joins of paths and cycles of equal order with sigma chromatic number 2

2021 ◽  
pp. 2250006
Author(s):  
Agnes D. Garciano ◽  
Maria Czarina T. Lagura ◽  
Reginaldo M. Marcelo
Keyword(s):  
Chromatic Number ◽  
Connected Graph ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Odd Cycle ◽  
Necessary And Sufficient ◽  
Positive Integers ◽  
Simple Connected Graph

For a simple connected graph [Formula: see text] let [Formula: see text] be a coloring of [Formula: see text] where two adjacent vertices may be assigned the same color. Let [Formula: see text] be the sum of colors of neighbors of any vertex [Formula: see text] The coloring [Formula: see text] is a sigma coloring of [Formula: see text] if for any two adjacent vertices [Formula: see text] [Formula: see text] The least number of colors required in a sigma coloring of [Formula: see text] is the sigma chromatic number of [Formula: see text] and is denoted by [Formula: see text] A sigma coloring of a graph is a neighbor-distinguishing type of coloring and it is known that the sigma chromatic number of a graph is bounded above by its chromatic number. It is also known that for a path [Formula: see text] and a cycle [Formula: see text] where [Formula: see text] [Formula: see text] and [Formula: see text] if [Formula: see text] is even. Let [Formula: see text] the join of the graphs [Formula: see text], where [Formula: see text] or [Formula: see text] [Formula: see text] and [Formula: see text] is not an odd cycle for any [Formula: see text]. It has been shown that if [Formula: see text] for [Formula: see text] and [Formula: see text] then [Formula: see text]. In this study, we give necessary and sufficient conditions under which [Formula: see text] where [Formula: see text] is the join of copies of [Formula: see text] and/or [Formula: see text] for the same value of [Formula: see text]. Let [Formula: see text] and [Formula: see text] be positive integers with [Formula: see text] and [Formula: see text] In this paper, we show that [Formula: see text] if and only if [Formula: see text] or [Formula: see text] is odd, [Formula: see text] is even and [Formula: see text]; and [Formula: see text] if and only if [Formula: see text] is even and [Formula: see text] We also obtain necessary and sufficient conditions on [Formula: see text] and [Formula: see text], so that [Formula: see text] for [Formula: see text] where [Formula: see text] or [Formula: see text] other than the cases [Formula: see text] and [Formula: see text]

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.

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