scholarly journals Hamiltonicity of Token Graphs of Some Join Graphs

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 1076
Author(s):  
Luis Enrique Adame ◽  
Luis Manuel Rivera ◽  
Ana Laura Trujillo-Negrete
Keyword(s):  
Infinite Family ◽  
Simple Graph ◽  
Symmetric Difference ◽  
Hamiltonian Graphs ◽  
Vertex Set

Let G be a simple graph of order n with vertex set V(G) and edge set E(G), and let k be an integer such that 1≤k≤n−1. The k-token graph G{k} of G is the graph whose vertices are the k-subsets of V(G), where two vertices A and B are adjacent in G{k} whenever their symmetric difference A▵B, defined as (A∖B)∪(B∖A), is a pair {a,b} of adjacent vertices in G. In this paper we study the Hamiltonicity of the k-token graphs of some join graphs. We provide an infinite family of graphs, containing Hamiltonian and non-Hamiltonian graphs, for which their k-token graphs are Hamiltonian. Our result provides, to our knowledge, the first family of non-Hamiltonian graphs for which it is proven the Hamiltonicity of their k-token graphs, for any 2<k<n−2.

Local dissymmetry on graphs and related algebraic structures

2019 ◽  
Vol 29 (08) ◽  
pp. 1499-1526 ◽  
Author(s):  
G. Chiaselotti ◽  
T. Gentile ◽  
F. Infusino
Keyword(s):  
Algebraic Structure ◽  
Binary Operation ◽  
Simple Graph ◽  
Symmetric Difference ◽  
Graph Structure ◽  
Algebraic Structures ◽  
Vertex Set ◽  
Structure Formula

We use the set symmetric difference between vertex subsets of a finite undirected simple graph [Formula: see text] to define a binary operation ∘ on the vertex set of a new graph [Formula: see text], that contains [Formula: see text] as subgraph and whose vertices are non-empty vertex subsets of [Formula: see text]. We show how the binary operation ∘ determines an algebraic structure on [Formula: see text] that is strictly related to the graph structure of [Formula: see text]. In fact, we show that [Formula: see text] agrees with [Formula: see text] and, next, we provide several characterizations for the algebraic structure [Formula: see text] when the graph [Formula: see text] is connected and locally dissymmetric.

scholarly journals Resistance Distance and Kirchhoff Index of Graphs with Pockets

2018 ◽  
Author(s):  
Qun Liu ◽  
Jiabao Liu
Keyword(s):  
Closed Form ◽  
Simple Graph ◽  
Kirchhoff Index ◽  
Resistance Distance ◽  
Vertex Set

Let G[F,Vk, Huv] be the graph with k pockets, where F is a simple graph of order n &ge; 1,Vk= {v1,v2,&middot;&middot;&middot; ,vk} is a subset of the vertex set of F and Hvis a simple graph of order m &ge; 2,v is a specified vertex of Hv. Also let G[F,Ek, Huv] be the graph with k edge pockets, where F is a simple graph of order n &ge; 2, Ek= {e1,e2,&middot;&middot;&middot;ek} is a subset of the edge set of F and Huvis a simple graph of order m &ge; 3, uv is a specified edge of Huvsuch that Huv&minus; u is isomorphic to Huv&minus; v. In this paper, we derive closed-form formulas for resistance distance and Kirchhoff index of G[F,Vk, Hv] and G[F,Ek, Huv] in terms of the resistance distance and Kirchhoff index F, Hv and F, Huv, respectively.

scholarly journals On k-rainbow domination in middle graphs

2021 ◽  
Author(s):  
Kijung Kim
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Simple Graph ◽  
Upper And Lower Bounds ◽  
Matching Number ◽  
Domatic Number ◽  
Rainbow Domination ◽  
Vertex Set ◽  
Middle Graph ◽  
Dominating Function

Let $G$ be a finite simple graph with vertex set $V(G)$ and edge set $E(G)$. A function $f : V(G) \rightarrow \mathcal{P}(\{1, 2, \dotsc, k\})$ is a \textit{$k$-rainbow dominating function} on $G$ if for each vertex $v \in V(G)$ for which $f(v)= \emptyset$, it holds that $\bigcup_{u \in N(v)}f(u) = \{1, 2, \dotsc, k\}$. The weight of a $k$-rainbow dominating function is the value $\sum_{v \in V(G)}|f(v)|$. The \textit{$k$-rainbow domination number} $\gamma_{rk}(G)$ is the minimum weight of a $k$-rainbow dominating function on $G$. In this paper, we initiate the study of $k$-rainbow domination numbers in middle graphs. We define the concept of a middle $k$-rainbow dominating function, obtain some bounds related to it and determine the middle $3$-rainbow domination number of some classes of graphs. We also provide upper and lower bounds for the middle $3$-rainbow domination number of trees in terms of the matching number. In addition, we determine the $3$-rainbow domatic number for the middle graph of paths and cycles.

scholarly journals Feedback Vertex Set on Hamiltonian Graphs

2021 ◽  
pp. 207-218
Author(s):  
Dario Cavallaro ◽  
Till Fluschnik
Keyword(s):  
Feedback Vertex Set ◽  
Hamiltonian Graphs ◽  
Vertex Set

scholarly journals On Total Vertex Irregularity Strength of Hexagonal Cluster Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Nurdin Hinding ◽  
Hye Kyung Kim ◽  
Nurtiti Sunusi ◽  
Riskawati Mise
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Simple Graph ◽  
Vertex Set ◽  
Cluster Graph ◽  
Cluster Graphs ◽  
Irregularity Strength

For a simple graph G with a vertex set V G and an edge set E G , a labeling f : V G ∪ ​ E G ⟶ 1,2 , ⋯ , k is called a vertex irregular total k − labeling of G if for any two different vertices x and y in V G we have w t x ≠ w t y where w t x = f x + ∑ u ∈ V G f x u . The smallest positive integer k such that G has a vertex irregular total k − labeling is called the total vertex irregularity strength of G , denoted by tvs G . The lower bound of tvs G for any graph G have been found by Baca et. al. In this paper, we determined the exact value of the total vertex irregularity strength of the hexagonal cluster graph on n cluster for n ≥ 2 . Moreover, we show that the total vertex irregularity strength of the hexagonal cluster graph on n cluster is 3 n 2 + 1 / 2 .

Walk-set induced connectedness in digital spaces

2017 ◽  
Vol 33 (2) ◽  
pp. 247-256
Author(s):  
JOSEF SLAPAL ◽  
Keyword(s):  
Jordan Curve ◽  
Digital Images ◽  
Simple Graph ◽  
Jordan Curve Theorem ◽  
Strong Product ◽  
Digital Plane ◽  
Vertex Set ◽  
Product Of Graphs ◽  
Digital Spaces

In an undirected simple graph, we define connectedness induced by a set of walks of the same lengths. We show that the connectedness is preserved by the strong product of graphs with walk sets. This result is used to introduce a graph on the vertex set Z2 with sets of walks that is obtained as the strong product of a pair of copies of a graph on the vertex set Z with certain walk sets. It is proved that each of the walk sets in the graph introduced induces connectedness on Z2 that satisfies a digital analogue of the Jordan curve theorem. It follows that the graph with any of the walk sets provides a convenient structure on the digital plane Z2 for the study of digital images.

On the edge irregularity strength of corona product of graphs with cycle

2020 ◽  
Vol 12 (06) ◽  
pp. 2050083
Author(s):  
I. Tarawneh ◽  
R. Hasni ◽  
A. Ahmad ◽  
G. C. Lau ◽  
S. M. Lee
Keyword(s):  
Simple Graph ◽  
Vertex Set ◽  
Product Of Graphs ◽  
Corona Product ◽  
Irregularity Strength

Let [Formula: see text] be a simple graph with vertex set [Formula: see text] and edge set [Formula: see text], respectively. An edge irregular [Formula: see text]-labeling of [Formula: see text] is a labeling of [Formula: see text] with labels from the set [Formula: see text] in such a way that for any two different edges [Formula: see text] and [Formula: see text], their weights [Formula: see text] and [Formula: see text] are distinct. The weight of an edge [Formula: see text] in [Formula: see text] is the sum of the labels of the end vertices [Formula: see text] and [Formula: see text]. The minimum [Formula: see text] for which the graph [Formula: see text] has an edge irregular [Formula: see text]-labeling is called the edge irregularity strength of [Formula: see text], denoted by [Formula: see text]. In this paper, we determine the exact value of edge irregularity strength of corona product of graphs with cycle.

scholarly journals Properties of the characteristic polynomials and spectrum of Pn and Cn

2016 ◽  
Vol 5 (2) ◽  
pp. 132
Author(s):  
Essam El Seidy ◽  
Salah Eldin Hussein ◽  
Atef Mohamed
Keyword(s):  
Adjacency Matrix ◽  
Laplacian Matrix ◽  
Simple Graph ◽  
Characteristic Polynomials ◽  
Signless Laplacian Matrix ◽  
Path Graph ◽  
Signless Laplacian ◽  
Vertex Set ◽  
Cycle Graph

We consider a finite undirected and connected simple graph  with vertex set  and edge set .We calculated the general formulas of the spectra of a cycle graph and path graph. In this discussion we are interested in the adjacency matrix, Laplacian matrix, signless Laplacian matrix, normalized Laplacian matrix, and seidel adjacency matrix.

When the annihilator-ideal graph is planar or complete bipartite graph

2019 ◽  
Vol 12 (02) ◽  
pp. 1950024
Author(s):  
M. J. Nikmehr ◽  
S. M. Hosseini
Keyword(s):  
Commutative Ring ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Simple Graph ◽  
Annihilator Ideal ◽  
Vertex Set

Let [Formula: see text] be a commutative ring with identity and [Formula: see text] be the set of ideals of [Formula: see text] with nonzero annihilator. The annihilator-ideal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with the vertex set [Formula: see text], and two distinct vertices [Formula: see text] and [Formula: see text] are adjacent if and only if [Formula: see text]. In this paper, we present some results on the bipartite, complete bipartite, outer planar and unicyclic of the annihilator-ideal graphs of a commutative ring. Among other results, bipartite annihilator-ideal graphs of rings are characterized. Also, we investigate planarity of the annihilator-ideal graph and classify rings whose annihilator-ideal graph is planar.

Sign in / Sign up

Export Citation Format

Share Document