On the Nullity Algorithm of Tree and Unicyclic Graph

2010 ◽  
pp. 48-55
Author(s):  
Tingzeng Wu ◽  
Defu Ma
Keyword(s):  
Unicyclic Graph

ALGEBRAIC CONNECTIVITY OF LOLLIPOP GRAPHS: A NEW APPROACH

2014 ◽  
Vol 06 (02) ◽  
pp. 1450028
Author(s):  
D. KALITA
Keyword(s):  
Linear Algebra ◽  
Unicyclic Graph ◽  
Algebraic Connectivity ◽  
New Approach ◽  
Pendent Vertex

The unicyclic graph Cn,gobtained by appending a cycle Cgof length g to a pendent vertex of a path on n - g vertices is the lollipop graph on n vertices. In [Algebraic connectivity of lollipop graphs, Linear Algebra Appl.434 (2011) 2204–2210], Guo et al. proved that a( Cn,g-1) < a( Cn,g) for g ≥ 4, where a( Cn,g) is the algebraic connectivity of Cn,g. In this paper, we present a new approach which is quite different from that of Guo et al. in proving a( Cn,g-1) < a( Cn,g) for g ≥ 4.

The Weakly Nilpotent Graph of a Commutative Ring

2017 ◽  
Vol 60 (2) ◽  
pp. 319-328
Author(s):  
Soheila Khojasteh ◽  
Mohammad Javad Nikmehr
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Disjoint Union ◽  
Independence Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Commutative Rings ◽  
Unicyclic Graph ◽  
Nilpotent Elements ◽  
Vertex Set

AbstractLet R be a commutative ring with non-zero identity. In this paper, we introduce theweakly nilpotent graph of a commutative ring. The weakly nilpotent graph of R denoted by Γw(R) is a graph with the vertex set R* and two vertices x and y are adjacent if and only if x y ∊ N(R)*, where R* = R \ {0} and N(R)* is the set of all non-zero nilpotent elements of R. In this article, we determine the diameter of weakly nilpotent graph of an Artinian ring. We prove that if Γw(R) is a forest, then Γw(R) is a union of a star and some isolated vertices. We study the clique number, the chromatic number, and the independence number of Γw(R). Among other results, we show that for an Artinian ring R, Γw(R) is not a disjoint union of cycles or a unicyclic graph. For Artinan rings, we determine diam . Finally, we characterize all commutative rings R for which is a cycle, where is the complement of the weakly nilpotent graph of R.

On the connected metric dimension of graphs and their complements

2020 ◽  
pp. 2150059
Author(s):  
Eunjeong Yi
Keyword(s):  
Unicyclic Graph ◽  
Metric Dimension ◽  
Resolving Set ◽  
Dimension Formula ◽  
Vertex Set

Let [Formula: see text] be a graph with vertex set [Formula: see text], and let [Formula: see text] denote the length of a shortest [Formula: see text] path in [Formula: see text]. A set [Formula: see text] is called a connected resolving set of [Formula: see text] if, for any distinct [Formula: see text], there exists a vertex [Formula: see text] such that [Formula: see text], and the subgraph of [Formula: see text] induced by [Formula: see text] is connected. The connected metric dimension, [Formula: see text], of [Formula: see text] is the minimum of the cardinalities over all connected resolving sets of [Formula: see text]. For a graph [Formula: see text] and its complement [Formula: see text], each of order [Formula: see text] and connected, we conjecture that [Formula: see text]; if [Formula: see text] is a tree or a unicyclic graph, we prove the conjecture and characterize graphs achieving equality.

scholarly journals An Approach to the Extremal Inverse Degree Index for Families of Graphs with Transformation Effect

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Muhammad Asif ◽  
Muhammad Hussain ◽  
Hamad Almohamedh ◽  
Khalid M. Alhamed ◽  
Sultan Almotairi
Keyword(s):  
Computer Program ◽  
Regular Graph ◽  
Connected Graph ◽  
Topological Index ◽  
Unicyclic Graph ◽  
Fixed Length ◽  
Inverse Degree ◽  
Fully Connected

The inverse degree index is a topological index first appeared as a conjuncture made by computer program Graffiti in 1988. In this work, we use transformations over graphs and characterize the inverse degree index for these transformed families of graphs. We established bonds for different families of n -vertex connected graph with pendent paths of fixed length attached with fully connected vertices under the effect of transformations applied on these paths. Moreover, we computed exact values of the inverse degree index for regular graph specifically unicyclic graph.

scholarly journals On the Estrada Indices of Unicyclic Graphs with Fixed Diameters

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2395
Author(s):  
Wenjie Ning ◽  
Kun Wang
Keyword(s):  
Adjacency Matrix ◽  
Connected Graph ◽  
Unicyclic Graph ◽  
Unicyclic Graphs ◽  
Estrada Index

The Estrada index of a graph G is defined as EE(G)=∑i=1neλi, where λ1,λ2,…,λn are the eigenvalues of the adjacency matrix of G. A unicyclic graph is a connected graph with a unique cycle. Let U(n,d) be the set of all unicyclic graphs with n vertices and diameter d. In this paper, we give some transformations which can be used to compare the Estrada indices of two graphs. Using these transformations, we determine the graphs with the maximum Estrada indices among U(n,d). We characterize two candidate graphs with the maximum Estrada index if d is odd and three candidate graphs with the maximum Estrada index if d is even.

scholarly journals Hamiltonian Cycles in Squares of Vertex-Unicyclic Graphs

1976 ◽  
Vol 19 (2) ◽  
pp. 169-172 ◽  
Author(s):  
Herbert Fleischner ◽  
Arthur M. Hobbs
Keyword(s):  
Sufficient Conditions ◽  
Unicyclic Graph ◽  
Necessary And Sufficient Conditions ◽  
Hamiltonian Cycles ◽  
Unicyclic Graphs ◽  
Necessary And Sufficient

In this paper we determine necessary and sufficient conditions for the square of a vertex-unicyclic graph to be Hamiltonian. The conditions are simple and easily checked. Further, we show that the square of a vertex-unicyclic graph is Hamiltonian if and only if it is vertex-pancyclic.

scholarly journals Unicyclic graphs with given number of cut vertices and the maximal Merrifield-Simmons index

Filomat ◽  
2014 ◽  
Vol 28 (3) ◽  
pp. 451-461 ◽  
Author(s):  
Hongbo Hua ◽  
Xinli Xu ◽  
Hongzhuan Wang
Keyword(s):  
Connected Graph ◽  
Independent Sets ◽  
Unicyclic Graph ◽  
Unicyclic Graphs

The Merrifield-Simmons index of a graph G, denoted by i(G), is defined to be the total number of independent sets in G, including the empty set. A connected graph is called a unicyclic graph, if it possesses equal number of vertices and edges. In this paper, we characterize the maximal unicyclic graph w.r.t. i(G) within all unicyclic graphs with given order and number of cut vertices. As a consequence, we determine the connected graph with at least one cycle, given number of cut vertices and the maximal Merrifield-Simmons index.

scholarly journals Unicyclic Graphs Whose Completely Regular Endomorphisms form a Monoid

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 240
Author(s):  
Rui Gu ◽  
Hailong Hou
Keyword(s):  
Sufficient Conditions ◽  
Unicyclic Graph ◽  
Necessary And Sufficient Conditions ◽  
Unicyclic Graphs ◽  
Completely Regular ◽  
Odd Cycle ◽  
Necessary And Sufficient

In this paper, completely regular endomorphisms of unicyclic graphs are explored. Let G be a unicyclic graph and let c E n d ( G ) be the set of all completely regular endomorphisms of G. The necessary and sufficient conditions under which c E n d ( G ) forms a monoid are given. It is shown that c E n d ( G ) forms a submonoid of E n d ( G ) if and only if G is an odd cycle or G = G ( n , m ) for some odd n ≥ 3 and integer m ≥ 1 .

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