Five results on maximizing topological indices in graphs

2021 ◽  
Vol vol. 23, no. 3 (Graph Theory) ◽  
Author(s):  
Stijn Cambie
Keyword(s):  
Independence Number ◽  
Wiener Index ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Matching Number ◽  
Extremal Graphs ◽  
Szeged Index ◽  
Complete Bipartite Graphs ◽  
The Difference

In this paper, we prove a collection of results on graphical indices. We determine the extremal graphs attaining the maximal generalized Wiener index (e.g. the hyper-Wiener index) among all graphs with given matching number or independence number. This generalizes some work of Dankelmann, as well as some work of Chung. We also show alternative proofs for two recents results on maximizing the Wiener index and external Wiener index by deriving it from earlier results. We end with proving two conjectures. We prove that the maximum for the difference of the Wiener index and the eccentricity is attained by the path if the order $n$ is at least $9$ and that the maximum weighted Szeged index of graphs of given order is attained by the balanced complete bipartite graphs.

On extremal bipartite graphs with given number of cut edges

2020 ◽  
Vol 12 (02) ◽  
pp. 2050015
Author(s):  
Hanlin Chen ◽  
Renfang Wu
Keyword(s):  
Topological Index ◽  
Wiener Index ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Extremal Graphs ◽  
Unified Approach ◽  
Harary Index ◽  
Eccentricity Index ◽  
Extremal Values

Let [Formula: see text] be a topological index of a graph. If [Formula: see text] (or [Formula: see text], respectively) for each edge [Formula: see text], then [Formula: see text] is monotonically decreasing (or increasing, respectively) with the addition of edges. In this paper, by a unified approach, we determine the extremal values of some monotonic topological indices, including the Wiener index, the hyper-Wiener index, the Harary index, the connective eccentricity index, the eccentricity distance sum, among all connected bipartite graphs with a given number of cut edges, and characterize the corresponding extremal graphs, respectively.

SOME RESULTS ON THE DIFFERENCE OF THE ZAGREB INDICES OF A GRAPH

2015 ◽  
Vol 92 (2) ◽  
pp. 177-186 ◽  
Author(s):  
MINGQIANG AN ◽  
LIMING XIONG
Keyword(s):  
Independence Number ◽  
Matching Number ◽  
Extremal Graphs ◽  
Vertex Connectivity ◽  
Zagreb Index ◽  
Sharp Bounds ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
The Difference ◽  
Appl Math

The classical first and second Zagreb indices of a graph $G$ are defined as $M_{1}(G)=\sum _{v\in V(G)}d(v)^{2}$ and $M_{2}(G)=\sum _{e=uv\in E(G)}d(u)d(v),$ where $d(v)$ is the degree of the vertex $v$ of $G.$ Recently, Furtula et al. [‘On difference of Zagreb indices’, Discrete Appl. Math.178 (2014), 83–88] studied the difference of $M_{1}$ and $M_{2},$ and showed that this difference is closely related to the vertex-degree-based invariant $RM_{2}(G)=\sum _{e=uv\in E(G)}[d(u)-1][d(v)-1]$, the reduced second Zagreb index. In this paper, we present sharp bounds for the reduced second Zagreb index, given the matching number, independence number and vertex connectivity, and we also completely determine the extremal graphs.

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 On the Wiener index and variants of the Szeged index of single-walled titania nanotubes TiO2(m,n)

2017 ◽  
Vol 95 (1) ◽  
pp. 68-86 ◽  
Author(s):  
Muhammad Imran ◽  
Sabeel-e Hafi
Keyword(s):  
Physicochemical Properties ◽  
Strain Energy ◽  
Chemical Compounds ◽  
Wiener Index ◽  
Exact Expression ◽  
Topological Indices ◽  
Titania Nanotubes ◽  
Graph Invariant ◽  
Szeged Index ◽  
Cut Method

Topological indices are numerical parameters of a graph that characterize its topology and are usually graph invariant. There are certain types of topological indices such as degree-based topological indices, distance-based topological indices, and counting-related topological indices. These topological indices correlate certain physicochemical properties such as boiling point, stability, and strain energy of chemical compounds. In this paper, we compute an exact expression of Wiener index, vertex-Szeged index, edge-Szeged index, and total-Szeged index of single-walled titania nanotubes TiO2(m, n) by using the cut method for all values of m and n.

scholarly journals On the difference between the revised Szeged index and the Wiener index

2014 ◽  
Vol 333 ◽  
pp. 28-34 ◽  
Author(s):  
Sandi Klavžar ◽  
M.J. Nadjafi-Arani
Keyword(s):  
Wiener Index ◽  
Szeged Index ◽  
The Difference

scholarly journals On extremal bipartite graphs with a given connectivity

Filomat ◽  
2019 ◽  
Vol 33 (6) ◽  
pp. 1531-1540
Author(s):  
Hanlin Chen ◽  
Hanyuan Deng ◽  
Renfang Wu
Keyword(s):  
Topological Index ◽  
Bipartite Graphs ◽  
Topological Indices ◽  
Extremal Graphs ◽  
Extremal Values

Let I(G) be a topological index of a graph. If I(G + e) < I(G) (or I(G + e) > I(G), respectively) for each edge e ? G, then I(G) is decreasing (or increasing, respectively) with addition of edges. In this paper, we determine the extremal values of some monotonic topological indices which decrease or increase with addition of edges, and characterize the corresponding extremal graphs among bipartite graphs with a given connectivity.

scholarly journals The Extremal Graphs of Some Topological Indices with Given Vertex k-Partiteness

Mathematics ◽  
2018 ◽  
Vol 6 (11) ◽  
pp. 271 ◽  
Author(s):  
Fang Gao ◽  
Xiaoxin Li ◽  
Kai Zhou ◽  
Jia-Bao Liu
Keyword(s):  
Wiener Index ◽  
Topological Indices ◽  
Extremal Graphs ◽  
Zeroth Order ◽  
First Zagreb Index ◽  
Zagreb Index ◽  
Randic Index ◽  
Laplacian Energy ◽  
Modified Wiener Index ◽  
Extremal Value

The vertex k-partiteness of graph G is defined as the fewest number of vertices whose deletion from G yields a k-partite graph. In this paper, we characterize the extremal value of the reformulated first Zagreb index, the multiplicative-sum Zagreb index, the general Laplacian-energy-like invariant, the general zeroth-order Randić index, and the modified-Wiener index among graphs of order n with vertex k-partiteness not more than m .

On the difference between the (revised) Szeged index and the Wiener index of cacti

2018 ◽  
Vol 247 ◽  
pp. 77-89 ◽  
Author(s):  
Sandi Klavžar ◽  
Shuchao Li ◽  
Huihui Zhang
Keyword(s):  
Wiener Index ◽  
Szeged Index ◽  
The Difference
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