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.

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 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 Signless Laplacian Spectral Radius of Unicyclic and Bicyclic Graphs with a Given Girth

10.37236/670 ◽  
2011 ◽  
Vol 18 (1) ◽  
Author(s):  
Ke Li ◽  
Ligong Wang ◽  
Guopeng Zhao
Keyword(s):  
Spectral Radius ◽  
Upper Bound ◽  
Extremal Graph ◽  
Laplacian Spectral Radius ◽  
Unicyclic Graphs ◽  
Signless Laplacian Spectral Radius ◽  
Disjoint Cycles ◽  
Signless Laplacian ◽  
Unique Graph ◽  
Bicyclic Graphs

Let $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$ be the set of unicyclic graphs and bicyclic graphs on $n$ vertices with girth $g$, respectively. Let $\mathcal{B}_{1}(n,g)$ be the subclass of $\mathcal{B}(n,g)$ consisting of all bicyclic graphs with two edge-disjoint cycles and $\mathcal{B}_{2}(n,g)=\mathcal{B}(n,g)\backslash\mathcal{B}_{1}(n,g)$. This paper determines the unique graph with the maximal signless Laplacian spectral radius among all graphs in $\mathcal{U}(n,g)$ and $\mathcal{B}(n,g)$, respectively. Furthermore, an upper bound of the signless Laplacian spectral radius and the extremal graph for $\mathcal{B}(n,g)$ are also given.

Unbalanced unicyclic and bicyclic graphs with extremal spectral radius

2020 ◽  
pp. 1-17
Author(s):  
Francesco Belardo ◽  
Maurizio Brunetti ◽  
Adriana Ciampella
Keyword(s):  
Spectral Radius ◽  
Bicyclic Graphs

Distance spectral radius of unicyclic graphs with fixed maximum degree

2019 ◽  
Vol 19 (04) ◽  
pp. 2050068
Author(s):  
Hezan Huang ◽  
Bo Zhou
Keyword(s):  
Spectral Radius ◽  
Connected Graph ◽  
Maximum Degree ◽  
Distance Matrix ◽  
The Other ◽  
Connected Graphs ◽  
Largest Eigenvalue ◽  
Unicyclic Graphs ◽  
The Largest Eigenvalue

The distance spectral radius of a connected graph is the largest eigenvalue of its distance matrix. For integers [Formula: see text] and [Formula: see text] with [Formula: see text], we prove that among the connected graphs on [Formula: see text] vertices of given maximum degree [Formula: see text] with at least one cycle, the graph [Formula: see text] uniquely maximizes the distance spectral radius, where [Formula: see text] is the graph obtained from the disjoint star on [Formula: see text] vertices and path on [Formula: see text] vertices by adding two edges, one connecting the star center with a path end, and the other being a chord of the star.

Signed graphs with small positive index of inertia

2016 ◽  
Vol 31 ◽  
pp. 232-243 ◽  
Author(s):  
Guihai Yu ◽  
Lihua Feng ◽  
Hui Qu
Keyword(s):  
Positive Eigenvalue ◽  
Signed Graphs ◽  
Unicyclic Graphs ◽  
Bicyclic Graphs ◽  
Positive Index

In this paper, the signed graphs with one positive eigenvalue are characterized, and the signed graphs with pendant vertices having exactly two positive eigenvalues are determined. As a consequence, the signed trees, the signed unicyclic graphs and the signed bicyclic graphs having one or two positive eigenvalues are characterized.

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