The harmonic index for unicyclic graphs with given girth

Filomat ◽  
2015 ◽  
Vol 29 (4) ◽  
pp. 673-686 ◽  
Author(s):  
Lingping Zhong ◽  
Qing Cui
Keyword(s):  
Extremal Graphs ◽  
Unicyclic Graphs ◽  
Harmonic Index ◽  
Degree Of A Vertex

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 a vertex u in G. In this work, we present the minimum, second-minimum, maximum and second-maximum harmonic indices for unicyclic graphs with given girth, and characterize the corresponding extremal graphs.

General sum-connectivity index of unicyclic graphs with given diameter and girth

2021 ◽  
pp. 2150140
Author(s):  
Tomáš Vetrík
Keyword(s):  
Lower Bounds ◽  
Topological Indices ◽  
Connectivity Index ◽  
Index Formula ◽  
Extremal Graphs ◽  
Unicyclic Graphs ◽  
Harmonic Index

Topological indices of graphs have been studied due to their extensive applications in chemistry. We obtain lower bounds on the general sum-connectivity index [Formula: see text] for unicyclic graphs [Formula: see text] of given girth and diameter, and for unicyclic graphs of given diameter, where [Formula: see text]. We present the extremal graphs for all the bounds. Our results generalize previously known results on the harmonic index for unicyclic graphs of given diameter.

The harmonic index of unicyclic graphs

2017 ◽  
Vol 10 (03) ◽  
pp. 1750039 ◽  
Author(s):  
R. Rasi ◽  
S. M. Sheikholeslami
Keyword(s):  
Maximum Degree ◽  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Harmonic Index ◽  
Degree Of A Vertex

The harmonic index of a graph [Formula: see text], denoted by [Formula: see text], is defined as the sum of weights [Formula: see text] over all edges [Formula: see text] of [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. Hu and Zhou [WSEAS Trans. Math. 12 (2013) 716–726] proved that for any unicyclic graph [Formula: see text] of order [Formula: see text], [Formula: see text] with equality if and only if [Formula: see text]. Recently, Zhong and Cui [Filomat 29 (2015) 673–686] generalized the above bound and proved that for any unicyclic graph [Formula: see text] of order [Formula: see text] other than [Formula: see text], [Formula: see text]. In this paper, we generalize the aforemention results and show that for any connected unicyclic graph [Formula: see text] of order [Formula: see text] with maximum degree [Formula: see text], [Formula: see text] and classify the extremal unicyclic graphs.

Maximum total irregularity index of some families of graph with maximum degree n − 1

2021 ◽  
pp. 2250069
Author(s):  
Shamaila Yousaf ◽  
Akhlaq Ahmad Bhatti
Keyword(s):  
Maximum Degree ◽  
Discrete Math ◽  
Simple Approach ◽  
Unicyclic Graphs ◽  
Irregularity Index ◽  
Degree Of A Vertex ◽  
Tricyclic Graphs ◽  
Bicyclic Graphs ◽  
Appl Math

The total irregularity index of a graph [Formula: see text] is defined by Abdo et al. [H. Abdo, S. Brandt and D. Dimitrov, The total irregularity of a graph, Discrete Math. Theor. Comput. Sci. 16 (2014) 201–206] as [Formula: see text], where [Formula: see text] denotes the degree of a vertex [Formula: see text]. In 2014, You et al. [L. H. You, J. S. Yang and Z. F. You, The maximal total irregularity of unicyclic graphs, Ars Comb. 114 (2014) 153–160.] characterized the graph having maximum [Formula: see text] value among all elements of the class [Formula: see text] (Unicyclic graphs) and Zhou et al. [L. H. You, J. S. Yang, Y. X. Zhu and Z. F. You, The maximal total irregularity of bicyclic graphs, J. Appl. Math. 2014 (2014) 785084, http://dx.doi.org/10.1155/2014/785084 ] characterized the graph having maximum [Formula: see text] value among all elements of the class [Formula: see text] (Bicyclic graphs). In this paper, we characterize the aforementioned graphs with an alternative but comparatively simple approach. Also, we characterized the graphs having maximum [Formula: see text] value among the classes [Formula: see text] (Tricyclic graphs), [Formula: see text] (Tetracyclic graphs), [Formula: see text] (Pentacyclic graphs) and [Formula: see text] (Hexacyclic graphs).

scholarly journals The Atom-Bond Connectivity Index of Catacondensed Polyomino Graphs

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Jinsong Chen ◽  
Jianping Liu ◽  
Qiaoliang Li
Keyword(s):  
Upper Bound ◽  
Connectivity Index ◽  
Zigzag Chain ◽  
Extremal Graphs ◽  
Degree Of A Vertex ◽  
Abc Index ◽  
Atom Bond

LetG=(V,E)be a graph. The atom-bond connectivity (ABC) index is defined as the sum of weights((du+dv−2)/dudv)1/2over all edgesuvofG, wheredudenotes the degree of a vertexuofG. In this paper, we give the atom-bond connectivity index of the zigzag chain polyomino graphs. Meanwhile, we obtain the sharp upper bound on the atom-bond connectivity index of catacondensed polyomino graphs withhsquares and determine the corresponding extremal graphs.

scholarly journals The harmonic index of unicyclic graphs with given matching number

2014 ◽  
Vol 38 (1) ◽  
pp. 173-183 ◽  
Author(s):  
L.V. Jian-Bo ◽  
Jianxi Li ◽  
Shiu Chee
Keyword(s):  
Matching Number ◽  
Unicyclic Graphs ◽  
Harmonic Index

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 variation of the Randic index with given girth and leaves

Filomat ◽  
2014 ◽  
Vol 28 (9) ◽  
pp. 1849-1853 ◽  
Author(s):  
Jianxi Liu
Keyword(s):  
Randić Index ◽  
Extremal Graphs ◽  
Connected Graphs ◽  
Randic Index ◽  
Minimum Value ◽  
Degree Of A Vertex

The variation of Randic index R'(G) of a graph G is defined by R'(G) = ?uv 1/ max{du,dv}, where du is the degree of a vertex u in G and the summation extends over all edges uv of G. In this work, we characterize the extremal trees achieving the minimum value of R0 for trees with given number of vertices and leaves. Furthermore, we characterize the extremal graphs achieving the minimum value of R' for connected graphs with given number of vertices and girth.

scholarly journals Spectral radius and extremal graphs for class of unicyclic graph with pendant vertices

2017 ◽  
Vol 9 (7) ◽  
pp. 168781401770713 ◽  
Author(s):  
Lu Zhi ◽  
Meijin Xu ◽  
Xiujuan Liu ◽  
Xiaodong Chen ◽  
Chen Chen ◽  
...  
Keyword(s):  
Spectral Radius ◽  
Upper Bounds ◽  
Unicyclic Graph ◽  
Extremal Graphs ◽  
Unicyclic Graphs

In this article, we research on the spectral radius of extremal graphs for the unicyclic graphs with girth g mainly by the graft transformation and matching and obtain the upper bounds of the spectral radius of unicyclic graphs.

scholarly journals The minimal total irregularity of some classes of graphs

Filomat ◽  
2016 ◽  
Vol 30 (5) ◽  
pp. 1203-1211 ◽  
Author(s):  
Yingxue Zhu ◽  
Lihua You ◽  
Jieshan Yang
Keyword(s):  
Open Problem ◽  
Vertex Degree ◽  
Connected Graphs ◽  
Unicyclic Graphs ◽  
The Third ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

In [1], Abdo and Dimitov defined the total irregularity of a graph G=(V,E) as irrt(G)=1/2 ?u,v?V|dG(u)-dG(v)|, where dG(u) denotes the vertex degree of a vertex u ? V. In this paper, we investigate the minimal total irregularity of the connected graphs, determine the minimal, the second minimal, the third minimal total irregularity of trees, unicyclic graphs, bicyclic graphs on n vertices, and propose an open problem for further research.

scholarly journals On Harmonic Index and Diameter of Quasi-Tree Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
A. Abdolghafourian ◽  
Mohammad A. Iranmanesh
Keyword(s):  
Lower Bounds ◽  
Tree Graph ◽  
Harmonic Index ◽  
Tree Graphs ◽  
Degree Of A Vertex

The harmonic index of a graph G ( H G ) is defined as the sum of the weights 2 / d u + d v for all edges u v of G , where d u is the degree of a vertex u in G . In this paper, we show that H G ≥ D G + 5 / 3 − n / 2 and H G ≥ 1 / 2 + 2 / 3 n − 2 D G , where G is a quasi-tree graph of order n and diameter D G . Indeed, we show that both lower bounds are tight and identify all quasi-tree graphs reaching these two lower bounds.

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