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 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.

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.

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 Ordering graphs with large eccentricity-based topological indices

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yunfang Tang ◽  
Xuli Qi
Keyword(s):  
Topological Indices ◽  
Extremal Graphs ◽  
Eccentric Connectivity Index ◽  
Unicyclic Graphs ◽  
Eccentricity Index ◽  
Universal Approach ◽  
Large Eccentricity ◽  
Bicyclic Graphs ◽  
Average Eccentricity ◽  
Appl Math

AbstractFor a connected graph, the first Zagreb eccentricity index $\xi _{1}$ ξ 1 is defined as the sum of the squares of the eccentricities of all vertices, and the second Zagreb eccentricity index $\xi _{2}$ ξ 2 is defined as the sum of the products of the eccentricities of pairs of adjacent vertices. In this paper, we mainly present a different and universal approach to determine the upper bounds respectively on the Zagreb eccentricity indices of trees, unicyclic graphs and bicyclic graphs, and characterize these corresponding extremal graphs, which extend the ordering results of trees, unicyclic graphs and bicyclic graphs in (Du et al. in Croat. Chem. Acta 85:359–362, 2012; Qi et al. in Discrete Appl. Math. 233:166–174, 2017; Li and Zhang in Appl. Math. Comput. 352:180–187, 2019). Specifically, we determine the n-vertex trees with the i-th largest indices $\xi _{1}$ ξ 1 and $\xi _{2}$ ξ 2 for i up to $\lfloor n/2+1 \rfloor $ ⌊ n / 2 + 1 ⌋ compared with the first three largest results of $\xi _{1}$ ξ 1 and $\xi _{2}$ ξ 2 in (Du et al. in Croat. Chem. Acta 85:359–362, 2012), the n-vertex unicyclic graphs with respectively the i-th and the j-th largest indices $\xi _{1}$ ξ 1 and $\xi _{2}$ ξ 2 for i up to $\lfloor n/2-1 \rfloor $ ⌊ n / 2 − 1 ⌋ and j up to $\lfloor 2n/5+1 \rfloor $ ⌊ 2 n / 5 + 1 ⌋ compared with respectively the first two and the first three largest results of $\xi _{1}$ ξ 1 and $\xi _{2}$ ξ 2 in (Qi et al. in Discrete Appl. Math. 233:166–174, 2017), and the n-vertex bicyclic graphs with respectively the i-th and the j-th largest indices $\xi _{1}$ ξ 1 and $\xi _{2}$ ξ 2 for i up to $\lfloor n/2-2\rfloor $ ⌊ n / 2 − 2 ⌋ and j up to $\lfloor 2n/15+1\rfloor $ ⌊ 2 n / 15 + 1 ⌋ compared with the first two largest results of $\xi _{2}$ ξ 2 in (Li and Zhang in Appl. Math. Comput. 352:180–187, 2019), where $n\ge 6$ n ≥ 6 . More importantly, we propose two kinds of index functions for the eccentricity-based topological indices, which can yield more general extremal results simultaneously for some classes of indices. As applications, we obtain and extend some ordering results about the average eccentricity of bicyclic graphs, and the eccentric connectivity index of trees, unicyclic graphs and bicyclic graphs.

SPECTRAL RADIUS AND AVERAGE 2-DEGREE SEQUENCE OF A GRAPH

2014 ◽  
Vol 06 (02) ◽  
pp. 1450029 ◽  
Author(s):  
YU-PEI HUANG ◽  
CHIH-WEN WENG
Keyword(s):  
Spectral Radius ◽  
Maximum Degree ◽  
Degree Sequence ◽  
Discrete Math ◽  
Upper Bounds ◽  
Laplacian Spectral Radius ◽  
Signless Laplacian Spectral Radius ◽  
Degree Of A Vertex ◽  
Signless Laplacian ◽  
Simple Connected Graph

In a simple connected graph, the average 2-degree of a vertex is the average degree of its neighbors. With the average 2-degree sequence and the maximum degree ratio of adjacent vertices, we present a sharp upper bound of the spectral radius of the adjacency matrix of a graph, which improves a result in [Y. H. Chen, R. Y. Pan and X. D. Zhang, Two sharp upper bounds for the signless Laplacian spectral radius of graphs, Discrete Math. Algorithms Appl.3(2) (2011) 185–191].

scholarly journals Ordering chemical graphs by Sombor indices and its applications

2021 ◽  
Vol 87 (01) ◽  
pp. 5-22
Author(s):  
Hechao Liu ◽  
◽  
Lihua You ◽  
Yufei Huang
Keyword(s):  
Chemical Properties ◽  
Topological Indices ◽  
Unicyclic Graphs ◽  
Degree Of Vertex ◽  
Numerical Invariants ◽  
Tricyclic Graphs ◽  
Bicyclic Graphs ◽  
Physical And Chemical ◽  
And Chemical Properties ◽  
Chemical Trees

Topological indices are a class of numerical invariants that predict certain physical and chemical properties of molecules. Recently, two novel topological indices, named as Sombor index and reduced Sombor index, were introduced by Gutman, defined as where denotes the degree of vertex in . In this paper, our aim is to order the chemical trees, chemical unicyclic graphs, chemical bicyclic graphs and chemical tricyclic graphs with respect to Sombor index and reduced Sombor index. We determine the first fourteen minimum chemical trees, the first four minimum chemical unicyclic graphs, the first three minimum chemical bicyclic graphs, the first seven minimum chemical tricyclic graphs. At last, we consider the applications of reduced Sombor index to octane isomers.

scholarly journals The A α -spectral radius of complements of bicyclic and tricyclic graphs with n vertices

2021 ◽  
Vol 10 (1) ◽  
pp. 56-66
Author(s):  
Chaohui Chen ◽  
Jiarong Peng ◽  
Tianyuan Chen
Keyword(s):  
Extremal Problem ◽  
Spectral Radius ◽  
Unicyclic Graphs ◽  
Tricyclic Graphs ◽  
Bicyclic Graphs

Abstract Recently, the extremal problem of the spectral radius in the class of complements of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs had been studied widely. In this paper, we extend the largest ordering of A α -spectral radius among all complements of bicyclic and tricyclic graphs with n vertices, respectively.

scholarly journals Bicyclic graphs with maximum degree resistance distance

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1625-1632 ◽  
Author(s):  
Junfeng Du ◽  
Jianhua Tu
Keyword(s):  
Maximum Degree ◽  
Minimum Degree ◽  
Theoretical Chemistry ◽  
Graph Invariants ◽  
Resistance Distance ◽  
Unicyclic Graphs ◽  
Degree Of Vertex ◽  
Bicyclic Graphs

Graph invariants, based on the distances between the vertices of a graph, are widely used in theoretical chemistry. Recently, Gutman, Feng and Yu (Transactions on Combinatorics, 01 (2012) 27- 40) introduced the degree resistance distance of a graph G, which is defined as DR(G) = ?{u,v}?V(G)[dG(u)+dG(v)]RG(u,v), where dG(u) is the degree of vertex u of the graph G, and RG(u, v) denotes the resistance distance between the vertices u and v of the graph G. Further, they characterized n-vertex unicyclic graphs having minimum and second minimum degree resistance distance. In this paper, we characterize n-vertex bicyclic graphs having maximum degree resistance distance.

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.

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