scholarly journals On the Construction of the Reflexive Vertex k-Labeling of Any Graph with Pendant Vertex

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
I. H. Agustin ◽  
M. I. Utoyo ◽  
Dafik ◽  
M. Venkatachalam ◽  
Surahmat
Keyword(s):  
Natural Number ◽  
Finite Graph ◽  
Star Graph ◽  
Pendant Vertex ◽  
Vertex Set

A total k-labeling is a function fe from the edge set to first natural number ke and a function fv from the vertex set to non negative even number up to 2kv, where k=maxke,2kv. A vertex irregular reflexivek-labeling of a simple, undirected, and finite graph G is total k-labeling, if for every two different vertices x and x′ of G, wtx≠wtx′, where wtx=fvx+Σxy∈EGfexy. The minimum k for graph G which has a vertex irregular reflexive k-labeling is called the reflexive vertex strength of the graph G, denoted by rvsG. In this paper, we determined the exact value of the reflexive vertex strength of any graph with pendant vertex which is useful to analyse the reflexive vertex strength on sunlet graph, helm graph, subdivided star graph, and broom graph.

Random flights on regular graphs

1984 ◽  
Vol 16 (03) ◽  
pp. 618-637 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Automorphism Group ◽  
Transition Probabilities ◽  
Finite Graph ◽  
Initial Position ◽  
Regular Graphs ◽  
Random Flights ◽  
The Past ◽  
Initial Location ◽  
Vertex Set

Let K be a finite graph with vertex set V = {x 0, x 1, …, xσ –1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by vn (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v 0 the initial location. It is assumed that the transition probabilities P{vn = xj | vn –1 = xi }, xi ϵ V, xj ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.

On permutability graphs of subgroups of groups

2015 ◽  
Vol 07 (02) ◽  
pp. 1550012 ◽  
Author(s):  
R. Rajkumar ◽  
P. Devi
Keyword(s):  
Finite Groups ◽  
The Other ◽  
Star Graph ◽  
Infinite Groups ◽  
Vertex Set

The permutability graph of subgroups of a given group G, denoted by Γ(G), is a graph with vertex set consists of all the proper subgroups of G and two distinct vertices in Γ(G) are adjacent if and only if the corresponding subgroups permute in G. In this paper, we classify the finite groups whose permutability graphs of subgroups are one of bipartite, star graph, C3-free, C5-free, K4-free, K5-free, K1,4-free, K2,3-free or Pn-free (n = 2, 3, 4). We investigate the same for infinite groups also. Moreover, some results on the girth, completeness and regularity of the permutability graphs of subgroups of groups are obtained. Among the other results, we characterize groups Q8, S3 and A4 by using their permutability graphs of subgroups.

scholarly journals BILANGAN KROMATIK LOKASI DARI GRAF P m P n ; K m P n ; DAN K , m K n

2013 ◽  
Vol 2 (1) ◽  
pp. 14
Author(s):  
Mariza Wenni
Keyword(s):  
Natural Number ◽  
Chromatic Number ◽  
Cartesian Product ◽  
Complete Graphs ◽  
Chromatic Numbers ◽  
Color Code ◽  
Connected Graphs ◽  
Vertex Set ◽  
Distinct Color

Let G and H be two connected graphs. Let c be a vertex k-coloring of aconnected graph G and let = fCg be a partition of V (G) into the resultingcolor classes. For each v 2 V (G), the color code of v is dened to be k-vector: c1; C2; :::; Ck(v) =(d(v; C1); d(v; C2); :::; d(v; Ck)), where d(v; Ci) = minfd(v; x) j x 2 Cg, 1 i k. Ifdistinct vertices have distinct color codes with respect to , then c is called a locatingcoloring of G. The locating chromatic number of G is the smallest natural number ksuch that there are locating coloring with k colors in G. The Cartesian product of graphG and H is a graph with vertex set V (G) V (H), where two vertices (a; b) and (a)are adjacent whenever a = a0and bb02 E(H), or aa0i2 E(G) and b = b, denotedby GH. In this paper, we will study about the locating chromatic numbers of thecartesian product of two paths, the cartesian product of paths and complete graphs, andthe cartesian product of two complete graphs.

scholarly journals More on the annihilator-ideal graph of a commutative ring

2019 ◽  
Vol 18 (08) ◽  
pp. 1950160
Author(s):  
M. J. Nikmehr ◽  
S. M. Hosseini
Keyword(s):  
Commutative Ring ◽  
Sufficient Conditions ◽  
Simple Graph ◽  
Necessary And Sufficient Conditions ◽  
Star Graph ◽  
Annihilator Ideal ◽  
Vertex Set ◽  
Necessary And Sufficient

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 study the affinity between the annihilator-ideal graph and the annihilating-ideal graph [Formula: see text] (a well known graph with the same vertices and two distinct vertices [Formula: see text] are adjacent if and only if [Formula: see text]) associated with [Formula: see text]. All rings whose [Formula: see text] and [Formula: see text] are characterized. Among other results, we obtain necessary and sufficient conditions under which [Formula: see text] is a star graph.

scholarly journals Edge irregular reflexive labeling for the $ r $-th power of the path

2021 ◽  
Vol 6 (10) ◽  
pp. 10405-10430
Author(s):  
Mohamed Basher ◽  
Keyword(s):  
Natural Number ◽  
Edge Strength ◽  
Vertex Set

<abstract><p>Let $ G(V, E) $ be a graph, where $ V(G) $ is the vertex set and $ E(G) $ is the edge set. Let $ k $ be a natural number, a total k-labeling $ \varphi:V(G)\bigcup E(G)\rightarrow \{0, 1, 2, 3, ..., k\} $ is called an edge irregular reflexive $ k $-labeling if the vertices of $ G $ are labeled with the set of even numbers from $ \{0, 1, 2, 3, ..., k\} $ and the edges of $ G $ are labeled with numbers from $ \{1, 2, 3, ..., k\} $ in such a way for every two different edges $ xy $ and $ x^{'}y^{'} $ their weights $ \varphi(x)+\varphi(xy)+\varphi(y) $ and $ \varphi(x^{'})+\varphi(x^{'}y^{'})+\varphi(y^{'}) $ are distinct. The reflexive edge strength of $ G $, $ res(G) $, is defined as the minimum $ k $ for which $ G $ has an edge irregular reflexive $ k $-labeling. In this paper, we determine the exact value of the reflexive edge strength for the $ r $-th power of the path $ P_{n} $, where $ r\geq2 $, $ n\geq r+4 $.</p></abstract>

scholarly journals Local Finiteness, Distinguishing Numbers, and Tucker's Conjecture

10.37236/4873 ◽  
2015 ◽  
Vol 22 (4) ◽  
Author(s):  
Florian Lehner ◽  
Rögnvaldur G. Möller
Keyword(s):  
Finite Number ◽  
Finite Graph ◽  
Locally Finite ◽  
Locally Finite Graph ◽  
Local Finiteness ◽  
Vertex Set

A distinguishing colouring of a graph is a colouring of the vertex set such that no non-trivial automorphism preserves the colouring. Tucker conjectured that if every non-trivial automorphism of a locally finite graph moves infinitely many vertices, then there is a distinguishing 2-colouring. We show that the requirement of local finiteness is necessary by giving a non-locally finite graph for which no finite number of colours suffices.

Random flights on regular graphs

1984 ◽  
Vol 16 (3) ◽  
pp. 618-637 ◽  
Author(s):  
Lajos Takács
Keyword(s):  
Automorphism Group ◽  
Transition Probabilities ◽  
Finite Graph ◽  
Initial Position ◽  
Regular Graphs ◽  
Random Flights ◽  
The Past ◽  
Initial Location ◽  
Vertex Set

Let K be a finite graph with vertex set V = {x0, x1, …, xσ–1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by vn (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v0 the initial location. It is assumed that the transition probabilities P{vn = xj | vn–1 = xi}, xi ϵ V, xj ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.

Bounds for signed double Roman k-domination in trees

2019 ◽  
Vol 53 (2) ◽  
pp. 627-643 ◽  
Author(s):  
Hong Yang ◽  
Pu Wu ◽  
Sakineh Nazari-Moghaddam ◽  
Seyed Mahmoud Sheikholeslami ◽  
Xiaosong Zhang ◽  
...  
Keyword(s):  
Minimum Weight ◽  
Domination Number ◽  
Finite Graph ◽  
Upper Bounds ◽  
Lower And Upper Bounds ◽  
Vertex Set ◽  
Dominating Function

Let k ≥ 1 be an integer and G be a simple and finite graph with vertex set V(G). A signed double Roman k-dominating function (SDRkDF) on a graph G is a function f:V(G) → {−1,1,2,3} such that (i) every vertex v with f(v) = −1 is adjacent to at least two vertices assigned a 2 or to at least one vertex w with f(w) = 3, (ii) every vertex v with f(v) = 1 is adjacent to at least one vertex w with f(w) ≥ 2 and (iii) ∑u∈N[v]f(u) ≥ k holds for any vertex v. The weight of a SDRkDF f is ∑u∈V(G) f(u), and the minimum weight of a SDRkDF is the signed double Roman k-domination number γksdR(G) of G. In this paper, we investigate the signed double Roman k-domination number of trees. In particular, we present lower and upper bounds on γksdR(T) for 2 ≤ k ≤ 6 and classify all extremal trees.

scholarly journals Distinguishability of Infinite Groups and Graphs

10.37236/2283 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Simon M Smith ◽  
Thomas W Tucker ◽  
Mark E Watkins
Keyword(s):  
Finite Graph ◽  
Primitive Permutation Group ◽  
Infinite Groups ◽  
Cartesian Products ◽  
Distinguishing Number ◽  
Locally Finite Graph ◽  
Point Stabilizer ◽  
Vertex Set ◽  
Vertex Transitive ◽  
Transitive Graphs

The distinguishing number of a group $G$ acting faithfully on a set $V$ is the least number of colors needed to color the elements of $V$ so that no non-identity element of the group preserves the coloring. The distinguishing number of a graph is the distinguishing number of its automorphism group acting on its vertex set. A connected graph $\Gamma$ is said to have connectivity 1 if there exists a vertex $\alpha \in V\Gamma$ such that $\Gamma \setminus \{\alpha\}$ is not connected. For $\alpha \in V$, an orbit of the point stabilizer $G_\alpha$ is called a suborbit of $G$.We prove that every nonnull, primitive graph with infinite diameter and countably many vertices has distinguishing number $2$. Consequently, any nonnull, infinite, primitive, locally finite graph is $2$-distinguishable; so, too, is any infinite primitive permutation group with finite suborbits. We also show that all denumerable vertex-transitive graphs of connectivity 1 and all Cartesian products of connected denumerable graphs of infinite diameter have distinguishing number $2$. All of our results follow directly from a versatile lemma which we call The Distinct Spheres Lemma.

scholarly journals Graph Homomorphisms between Trees

10.37236/4096 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Péter Csikvári ◽  
Zhicong Lin
Keyword(s):  
Lower Bound ◽  
Adjacency Matrix ◽  
Extremal Problems ◽  
Fixed Number ◽  
Star Graph ◽  
Spectral Entropy ◽  
Pendant Vertex ◽  
Path Graph ◽  
Graph Homomorphisms ◽  
Fixed Tree

In this paper we study several problems concerning the number of homomorphisms of trees. We begin with an algorithm for the number of homomorphisms from a tree to any graph. By using this algorithm and some transformations on trees, we study various extremal problems about the number of homomorphisms of trees. These applications include a far reaching generalization and a dual of Bollobás and Tyomkyn's result concerning the number of walks in trees.Some other main results of the paper are the following. Denote by $\hom(H,G)$ the number of homomorphisms from a graph $H$ to a graph $G$. For any tree $T_m$ on $m$ vertices we give a general lower bound for $\hom(T_m,G)$ by certain entropies of Markov chains defined on the graph $G$. As a particular case, we show that for any graph $G$, $$\exp(H_{\lambda}(G))\lambda^{m-1}\leq\hom(T_m,G),$$ where $\lambda$ is the largest eigenvalue of the adjacency matrix of $G$ and $H_{\lambda}(G)$ is a certain constant depending only on $G$ which we call the spectral entropy of $G$. We also show that if $T_m$ is any fixed tree and$$\hom(T_m,P_n)>\hom(T_m,T_n),$$for some tree $T_n$ on $n$ vertices, then $T_n$ must be the tree obtained from a path $P_{n-1}$ by attaching a pendant vertex to the second vertex of $P_{n-1}$.All the results together enable us to show that among all trees with fixed number of vertices, the path graph has the fewest number of endomorphisms while the star graph has the most.

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