scholarly journals The Maximal Total Irregularity of Bicyclic Graphs

2014 ◽  
Vol 2014 ◽  
pp. 1-9 ◽  
Author(s):  
Lihua You ◽  
Jieshan Yang ◽  
Yingxue Zhu ◽  
Zhifu You
Keyword(s):  
Vertex Degree ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

In 2012, Abdo and Dimitrov defined the total irregularity of a graphG=(V,E)asirrtG=1/2∑u,v∈VdGu-dGv, wheredGudenotes the vertex degree of a vertexu∈V. In this paper, we investigate the total irregularity of bicyclic graphs and characterize the graph with the maximal total irregularity among all bicyclic graphs onnvertices.

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.

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).

Sharp upper bounds of F-index among bicyclic graphs

2021 ◽  
pp. 2250055
Author(s):  
R. Khoeilar ◽  
A. Jahanbani ◽  
L. Shahbazi ◽  
J. Rodríguez
Keyword(s):  
Maximum Degree ◽  
Topological Index ◽  
Upper Bounds ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

The [Formula: see text]-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]. In this paper, we give sharp upper bounds of the [Formula: see text]-index (forgotten topological index) over bicyclic graphs, in terms of the order and maximum degree.

scholarly journals Bicyclic Graphs with the Second-Maximum and Third-Maximum Degree Resistance Distance

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Wenjie Ning ◽  
Kun Wang ◽  
Hassan Raza
Keyword(s):  
Connected Graph ◽  
Maximum Degree ◽  
Effective Resistance ◽  
Resistance Distance ◽  
Bicyclic Graph ◽  
Degree Of A Vertex ◽  
Bicyclic Graphs

Let G = V , E be a connected graph. The resistance distance between two vertices u and v in G , denoted by R G u , v , is the effective resistance between them if each edge of G is assumed to be a unit resistor. The degree resistance distance of G is defined as D R G = ∑ u , v ⊆ V G d G u + d G v R G u , v , where d G u is the degree of a vertex u in G and R G u , v is the resistance distance between u and v in G . A bicyclic graph is a connected graph G = V , E with E = V + 1 . This paper completely characterizes the graphs with the second-maximum and third-maximum degree resistance distance among all bicyclic graphs with n ≥ 6 vertices.

scholarly journals Two upper bounds for the degree distances of four sums of graphs

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 579-590 ◽  
Author(s):  
Mingqiang An ◽  
Liming Xiong ◽  
Kinkar Das
Keyword(s):  
Connected Graph ◽  
Wiener Index ◽  
Upper Bounds ◽  
Vertex Degree ◽  
Weighted Sum ◽  
Degree Of A Vertex ◽  
Degree Distance

The degree distance (DD), which is a weight version of the Wiener index, defined for a connected graph G as vertex-degree-weighted sum of the distances, that is, DD(G) = ?{u,v}?V(G)[dG(u)+dG(v)]d[u,v|G), where dG(u) denotes the degree of a vertex u in G and d(u,v|G) denotes the distance between two vertices u and v in G: In this paper, we establish two upper bounds for the degree distances of four sums of two graphs in terms of other indices of two individual graphs.

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 New upper bounds for the forgotten index among bicyclic graphs

2020 ◽  
Author(s):  
Jonnathan Rodriguez ◽  
Akbar Jahanbani ◽  
Seyed Mahmoud Sheikholeslami ◽  
Reza Rasi ◽  
L Shahbazi
Keyword(s):  
Maximum Degree ◽  
Topological Index ◽  
Upper Bounds ◽  
Degree Of A Vertex ◽  
Forgotten Index ◽  
Bicyclic Graphs

The forgotten topological index of a graph $G$, denoted by $F(G)$, is defined as the sum of weights $d(u)^{2}+d(v)^{2}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$. In this paper, we give sharp upper bounds of the F-index (forgotten topological index) over bicyclic graphs, in terms of the order and maximum degree.

scholarly journals The node cop‐win reliability of unicyclic and bicyclic graphs

Networks ◽  
2021 ◽  
Author(s):  
Maimoonah Ahmed ◽  
Ben Cameron
Keyword(s):  
Bicyclic Graphs
Sign in / Sign up

Export Citation Format

Share Document