scholarly journals A lower bound for the harmonic index of a graph with minimum degree at least three

Filomat ◽  
2016 ◽  
Vol 30 (8) ◽  
pp. 2249-2260 ◽  
Author(s):  
Minghong Cheng ◽  
Ligong Wang
Keyword(s):  
Lower Bound ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Harmonic Index ◽  
Degree Of A Vertex

The harmonic index H(G) of a graph G is the sum of the weights 2/d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this work, a lower bound for the harmonic index of a graph with minimum degree at least three is obtained and the corresponding extremal graph is characterized.

scholarly journals On the harmonic index of bicyclic conjugated molecular graphs

Filomat ◽  
2014 ◽  
Vol 28 (2) ◽  
pp. 421-428 ◽  
Author(s):  
Yan Zhu ◽  
Renying Chang
Keyword(s):  
Lower Bound ◽  
Perfect Matching ◽  
Harmonic Index ◽  
Molecular Graphs ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs ◽  
Sharp Lower Bound ◽  
Given Size Of Matching

The harmonic index H(G) of a graph G is defined as the sum of weights 2/ d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we first present a sharp lower bound on the harmonic index of bicyclic conjugated molecular graphs (bicyclic graphs with perfect matching). Also a sharp lower bound on the harmonic index of bicyclic graphs is given in terms of the order and given size of matching.

scholarly journals A lower bound for the harmonic index of a graph with minimum degree at least two

Filomat ◽  
2013 ◽  
Vol 27 (1) ◽  
pp. 51-55 ◽  
Author(s):  
Renfang Wu ◽  
Zikai Tang ◽  
Hanyuan Deng
Keyword(s):  
Lower Bound ◽  
Minimum Degree ◽  
Harmonic Index

The minimum value of the harmonic index for a graph with the minimum degree two

2018 ◽  
Vol 13 (03) ◽  
pp. 2050054 ◽  
Author(s):  
Hanyuan Deng ◽  
S. Balachandran ◽  
S. Raja Balachandar
Keyword(s):  
Linear Programming ◽  
Connected Graph ◽  
Minimum Degree ◽  
Harmonic Index ◽  
Minimum Value ◽  
Degree Of A Vertex

For a simple and connected graph [Formula: see text] with [Formula: see text] vertices and the minimum degree two, we show that [Formula: see text] by a technique based on linear programming, where [Formula: see text] is the harmonic index of a graph [Formula: see text], defined as the sum of the weights [Formula: see text] of all edges [Formula: see text] of [Formula: see text], [Formula: see text] denotes the degree of a vertex [Formula: see text], and characterize the graph with the minimum value.

Counter examples to a conjecture concerning harmonic index

2018 ◽  
Vol 11 (03) ◽  
pp. 1850035 ◽  
Author(s):  
Akbar Ali
Keyword(s):  
Lower Bound ◽  
Minimum Degree ◽  
Harmonic Index

A recently proposed conjecture by Cheng and Wang [A lower bound for the harmonic index of a graph with minimum degree at least three, Filomat 30(8) (2016) 2249–2260] about the harmonic index is disproved.

On the distance signless Laplacian spectral radius of graphs and digraphs

2017 ◽  
Vol 32 ◽  
pp. 438-446 ◽  
Author(s):  
Dan Li ◽  
Guoping Wang ◽  
Jixiang Meng
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Minimal Distance ◽  
Vertex Connectivity ◽  
Laplacian Spectral Radius ◽  
Connected Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Signless Laplacian

Let \eta(G) denote the distance signless Laplacian spectral radius of a connected graph G. In this paper,bounds for the distance signless Laplacian spectral radius of connected graphs are given, and the extremal graph with the minimal distance signless Laplacian spectral radius among the graphs with given vertex connectivity and minimum degree is determined. Furthermore, the digraph that minimizes the distance signless Laplacian spectral radius with given vertex connectivity is characterized.

scholarly journals On the sum-connectivity index

Filomat ◽  
2011 ◽  
Vol 25 (3) ◽  
pp. 29-42 ◽  
Author(s):  
Shilin Wang ◽  
Zhou Bo ◽  
Nenad Trinajstic
Keyword(s):  
Lower Bound ◽  
Minimum Degree ◽  
Simple Graph ◽  
Connectivity Index ◽  
Extremal Graphs ◽  
Free Graph ◽  
Mathematical Chemistry ◽  
Degree Of Vertex

The sum-connectivity index of a simple graph G is defined in mathematical chemistry as R+(G) = ? uv?E(G)(du+dv)?1/2, where E(G) is the edge set of G and du is the degree of vertex u in G. We give a best possible lower bound for the sum-connectivity index of a graph (a triangle-free graph, respectively) with n vertices and minimum degree at least two and characterize the extremal graphs, where n ? 11.

scholarly journals Extensions of the Erdős–Gallai theorem and Luo’s theorem

2019 ◽  
Vol 29 (1) ◽  
pp. 128-136 ◽  
Author(s):  
Bo Ning ◽  
Xing Peng
Keyword(s):  
Graph Theory ◽  
Upper Bound ◽  
Connected Graph ◽  
Minimum Degree ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Classical Theorem ◽  
Vertex Graph ◽  
Turán Number ◽  
Image Position

AbstractThe famous Erdős–Gallai theorem on the Turán number of paths states that every graph with n vertices and m edges contains a path with at least (2m)/n edges. In this note, we first establish a simple but novel extension of the Erdős–Gallai theorem by proving that every graph G contains a path with at least $${{(s + 1){N_{s + 1}}(G)} \over {{N_s}(G)}} + s - 1$$ edges, where Nj(G) denotes the number of j-cliques in G for 1≤ j ≤ ω(G). We also construct a family of graphs which shows our extension improves the estimate given by the Erdős–Gallai theorem. Among applications, we show, for example, that the main results of [20], which are on the maximum possible number of s-cliques in an n-vertex graph without a path with ℓ vertices (and without cycles of length at least c), can be easily deduced from this extension. Indeed, to prove these results, Luo [20] generalized a classical theorem of Kopylov and established a tight upper bound on the number of s-cliques in an n-vertex 2-connected graph with circumference less than c. We prove a similar result for an n-vertex 2-connected graph with circumference less than c and large minimum degree. We conclude this paper with an application of our results to a problem from spectral extremal graph theory on consecutive lengths of cycles in graphs.

scholarly journals Triangle-Free Subgraphs of Random Graphs

2017 ◽  
Vol 27 (2) ◽  
pp. 141-161
Author(s):  
PETER ALLEN ◽  
JULIA BÖTTCHER ◽  
YOSHIHARU KOHAYAKAWA ◽  
BARNABY ROBERTS
Keyword(s):  
Graph Theory ◽  
Random Graph ◽  
Random Graphs ◽  
Minimum Degree ◽  
Discrete Math ◽  
Constant Factor ◽  
Extremal Graph ◽  
Extremal Graph Theory ◽  
Spanning Subgraph

Recently there has been much interest in studying random graph analogues of well-known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of G(n, p) with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of G(n, p) with minimum degree at least (2/5 + o(1))pn is (p−1n)-close to bipartite, and each spanning triangle-free subgraph of G(n, p) with minimum degree at least (1/3 + ϵ)pn is O(p−1n)-close to r-partite for some r = r(ϵ). These are random graph analogues of a result by Andrásfai, Erdős and Sós (Discrete Math.8 (1974), 205–218), and a result by Thomassen (Combinatorica22 (2002), 591–596). We also show that our results are best possible up to a constant factor.

scholarly journals $\delta$-Connectivity in Random Lifts of Graphs

10.37236/6639 ◽  
2017 ◽  
Vol 24 (1) ◽  
Author(s):  
Shashwat Silas
Keyword(s):  
Lower Bound ◽  
Symmetric Group ◽  
Wreath Product ◽  
Connected Graph ◽  
Minimum Degree ◽  
Symmetric Groups ◽  
Gamma Delta ◽  
Random Generators

Amit and Linial have shown that a random lift of a connected graph with minimum degree $\delta\ge3$ is asymptotically almost surely (a.a.s.) $\delta$-connected and mentioned the problem of estimating this probability as a function of the degree of the lift. Using a connection between a random $n$-lift of a graph and a randomly generated subgroup of the symmetric group on $n$-elements, we show that this probability is at least  $1 - O\left(\frac{1}{n^{\gamma(\delta)}}\right)$ where $\gamma(\delta)>0$ for $\delta\ge 5$ and it is strictly increasing with $\delta$. We extend this to show that one may allow $\delta$ to grow slowly as a function of the degree of the lift and the number of vertices and still obtain that random lifts are a.a.s. $\delta$-connected. We also simplify a later result showing a lower bound on the edge expansion of random lifts. On a related note, we calculate the probability that a subgroup of a wreath product of symmetric groups generated by random generators is transitive, extending a well known result of Dixon which covers the case for subgroups of the symmetric group.

scholarly journals On a conjecture of the harmonic index and the minimum degree of graphs

Filomat ◽  
2018 ◽  
Vol 32 (10) ◽  
pp. 3435-3441 ◽  
Author(s):  
Xiaoling Sun ◽  
Yubin Gao ◽  
Jianwei Du ◽  
Lan Xu
Keyword(s):  
Minimum Degree ◽  
Split Graph ◽  
Connected Graphs ◽  
Harmonic Index

The harmonic index of a graph G is defined as the sum of the weights 2/ d(u)+d(v) of all edges uv of G, where d(u) denotes the degree of the vertex u in G. Cheng and Wang [4] proposed a conjecture: For all connected graphs G with n ? 4 vertices and minimum degree ?(G) ? k, where 1 ? k ? ?n2?+ 1, then H(G) ? H(K*k,n-k) with equality if and only if G ? K*k,n-k. K*k,n-k is a complete split graph which has only two degrees, i.e. degree k and degree n-1, and the number of vertices of degree k is n-k, while the number of vertices of degree n-1 is k. In this work, we prove that this conjecture is true when k ? n2, and give a counterexample to show that the conjecture is not correct when k = ?n/2? + 1, n is even, that is k = n/2 + 1.

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