scholarly journals On symmetric division deg index of unicyclic graphs and bicyclic graphs with given matching number

2021 ◽  
Vol 6 (8) ◽  
pp. 9020-9035
Author(s):  
Xiaoling Sun ◽  
◽  
Yubin Gao ◽  
Jianwei Du
Keyword(s):  
Matching Number ◽  
Unicyclic Graphs ◽  
Symmetric Division ◽  
Bicyclic 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 Regularity of bicyclic graphs and their powers

2019 ◽  
Vol 19 (03) ◽  
pp. 2050057 ◽  
Author(s):  
Yairon Cid-Ruiz ◽  
Sepehr Jafari ◽  
Navid Nemati ◽  
Beatrice Picone
Keyword(s):  
Base Case ◽  
Matching Number ◽  
Edge Ideal ◽  
Induced Matching ◽  
Base Graph ◽  
Bicyclic Graph ◽  
Bicyclic Graphs

Let [Formula: see text] be the edge ideal of a bicyclic graph [Formula: see text] with a dumbbell as the base graph. In this paper, we characterize the Castelnuovo–Mumford regularity of [Formula: see text] in terms of the induced matching number of [Formula: see text]. For the base case of this family of graphs, i.e. dumbbell graphs, we explicitly compute the induced matching number. Moreover, we prove that [Formula: see text], for all [Formula: see text], when [Formula: see text] is a dumbbell graph with a connecting path having no more than two vertices.

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.

scholarly journals On the nullity and the matching number of unicyclic graphs

2009 ◽  
Vol 431 (8) ◽  
pp. 1293-1301 ◽  
Author(s):  
Ji-Ming Guo ◽  
Weigen Yan ◽  
Yeong-Nan Yeh
Keyword(s):  
Matching Number ◽  
Unicyclic 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 Locating-Total Domination Number of Cacti Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-10
Author(s):  
Jianxin Wei ◽  
Uzma Ahmad ◽  
Saira Hameed ◽  
Javaria Hanif
Keyword(s):  
Upper Bound ◽  
Connected Graph ◽  
Dominating Set ◽  
Domination Number ◽  
Total Domination ◽  
Total Domination Number ◽  
Unicyclic Graphs ◽  
Total Dominating Set ◽  
Bicyclic Graphs ◽  
Number Of Cycles

For a connected graph J, a subset W ⊆ V J is termed as a locating-total dominating set if for a ∈ V J ,   N a ∩ W ≠ ϕ , and for a ,   b ∈ V J − W ,   N a ∩ W ≠ N b ∩ W . The number of elements in a smallest such subset is termed as the locating-total domination number of J. In this paper, the locating-total domination number of unicyclic graphs and bicyclic graphs are studied and their bounds are presented. Then, by using these bounds, an upper bound for cacti graphs in terms of their order and number of cycles is estimated. Moreover, the exact values of this domination variant for some families of cacti graphs including tadpole graphs and rooted products are also determined.

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