Valued Field Graph and Some Related Parameters

2020 ◽  
pp. 389-395
Author(s):  
G. Suresh Singh ◽  
P. K. Prasobha
Keyword(s):  
Finite Field ◽  
Independence Number ◽  
Valued Field ◽  
Vertex Set ◽  
Adic Valuation

Let $K$ be any finite field. For any prime $p$, the $p$-adic valuation map is given by $\psi_{p}:K/\{0\} \to \R^+\bigcup\{0\}$ is given by $\psi_{p}(r) = n$ where $r = p^n \frac{a}{b}$, where $p,a,b$ are relatively prime. The field $K$ together with a valuation is called valued field. Also, any field $K$ has the trivial valuation determined by $\psi{(K)} = \{0,1\}$. Through out the paper K represents $\Z_q$. In this paper, we construct the graph corresponding to the valuation map called the valued field graph, denoted by $VFG_{p}(\Z_{q})$ whose vertex set is $\{v_0,v_1,v_2,\ldots, v_{q-1}\}$ where two vertices $v_i$ and $v_j$ are adjacent if $\psi_{p}(i) = j$ or $\psi_{p}(j) = i$. Here, we tried to characterize the valued field graph in $\Z_q$. Also we analyse various graph theoretical parameters such as diameter, independence number etc.

scholarly journals On the Unitary Cayley Graph of a Ring

10.37236/2214 ◽  
2012 ◽  
Vol 19 (2) ◽  
Author(s):  
Dariush Kiani ◽  
Mohsen Molla Haji Aghaei
Keyword(s):  
Finite Field ◽  
Cayley Graph ◽  
Independence Number ◽  
Commutative Rings ◽  
Finite Ring ◽  
Vertex Set ◽  
Unitary Cayley Graph ◽  
Finite Commutative Rings

Let $R$ be a ring with identity. The unitary Cayley graph of a ring $R$, denoted by $G_{R}$, is the graph, whose vertex set is $R$, and in which $\{x,y\}$ is an edge if and only if $x-y$ is a unit of $R$. In this paper we find chromatic, clique and independence number of $G_{R}$, where $R$ is a finite ring. Also, we prove that if $G_{R} \simeq G_{S}$, then $G_{R/J_{R}} \simeq G_{S/J_{S}}$, where $\rm J_{R}$ and $\rm J_{S}$ are Jacobson radicals of $R$ and $S$, respectively. Moreover, we prove if $G_{R} \simeq G_{M_{n}(F)}$ then $R\simeq M_{n}(F)$, where $R$ is a ring and $F$ is a finite field. Finally, let $R$ and $S$ be finite commutative rings, we show that if $G_{R} \simeq G_{S}$, then $\rm R/ {J}_{R}\simeq S/J_{S}$.

The genus of graphs associated with vector spaces

2019 ◽  
Vol 19 (05) ◽  
pp. 2050086 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
K. Prabha Ananthi
Keyword(s):  
Finite Field ◽  
Vector Space ◽  
Undirected Graph ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Vertex Connectivity ◽  
Dimensional Vector Space ◽  
Nonzero Component ◽  
Vertex Set ◽  
Component Graph

Let [Formula: see text] be a k-dimensional vector space over a finite field [Formula: see text] with a basis [Formula: see text]. The nonzero component graph of [Formula: see text], denoted by [Formula: see text], is a simple undirected graph with vertex set as nonzero vectors of [Formula: see text] such that there is an edge between two distinct vertices [Formula: see text] if and only if there exists at least one [Formula: see text] along which both [Formula: see text] and [Formula: see text] have nonzero scalars. In this paper, we find the vertex connectivity and girth of [Formula: see text]. We also characterize all vector spaces [Formula: see text] for which [Formula: see text] has genus either 0 or 1 or 2.

THE INTERSECTION GRAPH OF GAMMA SETS IN THE TOTAL GRAPH OF A COMMUTATIVE RING-II

2013 ◽  
Vol 12 (04) ◽  
pp. 1250199 ◽  
Author(s):  
T. ASIR ◽  
T. TAMIZH CHELVAM
Keyword(s):  
Commutative Ring ◽  
Independence Number ◽  
Clique Number ◽  
Commutative Rings ◽  
Intersection Graph ◽  
Vertex Connectivity ◽  
Total Graph ◽  
Graph Theoretic ◽  
Vertex Set ◽  
Finite Commutative Rings

The intersection graph ITΓ(R) of gamma sets in the total graph TΓ(R) of a commutative ring R, is the undirected graph with vertex set as the collection of all γ-sets in the total graph of R and two distinct vertices u and v are adjacent if and only if u ∩ v ≠ ∅. Tamizh Chelvam and Asir [The intersection graph of gamma sets in the total graph I, to appear in J. Algebra Appl.] studied about ITΓ(R) where R is a commutative Artin ring. In this paper, we continue our interest on ITΓ(R) and actually we study about Eulerian, Hamiltonian and pancyclic nature of ITΓ(R). Further, we focus on certain graph theoretic parameters of ITΓ(R) like the independence number, the clique number and the connectivity of ITΓ(R). Also, we obtain both vertex and edge chromatic numbers of ITΓ(R). In fact, it is proved that if R is a finite commutative ring, then χ(ITΓ(R)) = ω(ITΓ(R)). Having proved that ITΓ(R) is weakly perfect for all finite commutative rings, we further characterize all finite commutative rings for which ITΓ(R) is perfect. In this sequel, we characterize all commutative Artin rings for which ITΓ(R) is of class one (i.e. χ′(ITΓ(R)) = Δ(ITΓ(R))). Finally, it is proved that the vertex connectivity and edge connectivity of ITΓ(R) are equal to the degree of any vertex in ITΓ(R).

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.

THE JACOBSON GRAPH OF COMMUTATIVE RINGS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250179 ◽  
Author(s):  
A. AZIMI ◽  
A. ERFANIAN ◽  
M. FARROKHI D. G.
Keyword(s):  
Commutative Ring ◽  
Chromatic Number ◽  
Independence Number ◽  
Commutative Rings ◽  
Dominating Number ◽  
Jacobson Graph ◽  
Vertex Set ◽  
Numerical Invariants ◽  
Nonzero Identity

Let R be a commutative ring with nonzero identity. Then the Jacobson graph of R, denoted by 𝔍R, is defined as a graph with vertex set R\J(R) such that two distinct vertices x and y are adjacent if and only if 1 - xy is not a unit of R. We obtain some graph theoretical properties of 𝔍R including its connectivity, planarity and perfectness and we compute some of its numerical invariants, namely diameter, girth, dominating number, independence number and vertex chromatic number and give an estimate for its edge chromatic number.

scholarly journals Signed total k-independence in digraphs

Filomat ◽  
2014 ◽  
Vol 28 (10) ◽  
pp. 2121-2130
Author(s):  
Lutz Volkmann
Keyword(s):  
Independence Number ◽  
Upper Bounds ◽  
Vertex Set

Let k ? 2 be an integer. A function f:V(D) ? {-1,1} defined on the vertex set V(D) of a digraph D is a signed total k-independence function if ?x?N-(v)f(x) ? k - 1 for each v ? V(D), where N-(v) consists of all vertices of D from which arcs go into v. The weight of a signed total k-independence function f is defined by w(f)=?x?V(D)f(x). The maximum of weights w(f), taken over all signed total k-independence functions f on D, is the signed total k-independence number k?st(D) of D. In this work, we mainly present upper bounds on k?st(D), as for example k?st(D) ? n-2? ?- + 1-k)/2? and k?st(D)? ?+2k-?+-2/?+?+ ? n , where n is the order, ?- the maximum indegree and ?+ and ?+ are the maximum and minimum outdegree of the digraph D. Some of our results imply well-known properties on the signed total 2-independence number of graphs.

THE ITERATION DIGRAPHS OF GROUP RINGS OVER FINITE FIELDS

2014 ◽  
Vol 13 (05) ◽  
pp. 1350162 ◽  
Author(s):  
YANGJIANG WEI ◽  
GAOHUA TANG ◽  
JIZHU NAN
Keyword(s):  
Positive Integer ◽  
Finite Field ◽  
Finite Fields ◽  
Sufficient Conditions ◽  
Group Rings ◽  
Directed Edge ◽  
Cycle Structure ◽  
Vertex Set ◽  
Finite Commutative Ring ◽  
Necessary And Sufficient

For a finite commutative ring R and a positive integer k ≥ 2, we construct an iteration digraph G(R, k) whose vertex set is R and for which there is a directed edge from a ∈ R to b ∈ R if b = ak. In this paper, we investigate the iteration digraphs G(𝔽prCn, k) of 𝔽prCn, the group ring of a cyclic group Cn over a finite field 𝔽pr. We study the cycle structure of G(𝔽prCn, k), and explore the symmetric digraphs. Finally, we obtain necessary and sufficient conditions on 𝔽prCn and k such that G(𝔽prCn, k) is semiregular.

scholarly journals The Non-Crossing Graph

10.37236/1140 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Nathan Linial ◽  
Michael Saks ◽  
David Statter
Keyword(s):  
Chromatic Number ◽  
Independence Number ◽  
Clique Number ◽  
The Other ◽  
Fractional Chromatic Number ◽  
Vertex Set

Two sets are non-crossing if they are disjoint or one contains the other. The non-crossing graph ${\rm NC}_n$ is the graph whose vertex set is the set of nonempty subsets of $[n]=\{1,\ldots,n\}$ with an edge between any two non-crossing sets. Various facts, some new and some already known, concerning the chromatic number, fractional chromatic number, independence number, clique number and clique cover number of this graph are presented. For the chromatic number of this graph we show: $$ n(\log_e n -\Theta(1)) \le \chi({\rm NC}_n) \le n (\lceil\log_2 n\rceil-1). $$

scholarly journals Monochromatic Loose-Cycle Partitions in Hypergraphs

10.37236/4062 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
András Gyárfás ◽  
Gábor Sárközy
Keyword(s):  
Independence Number ◽  
Uniform Hypergraph ◽  
Partition Problem ◽  
Vertex Set

In this paper we study the monochromatic loose-cycle partition problem for non-complete hypergraphs. Our main result is that in any $r$-coloring of a $k$-uniform hypergraph with independence number $\alpha$ there is a partition of the vertex set into monochromatic loose cycles such that their number depends only on $r$, $k$ and $\alpha$. We also give an extension of the following result of Pósa to hypergraphs: the vertex set of every graph $G$ can be partitioned into at most $\alpha(G)$ cycles, edges and vertices.

scholarly journals Nonessential sum graph of an Artinian ring

2021 ◽  
Vol ahead-of-print (ahead-of-print) ◽  
Author(s):  
Bikash Barman ◽  
Kukil Kalpa Rajkhowa
Keyword(s):  
Design Methodology ◽  
Undirected Graph ◽  
Independence Number ◽  
Domination Number ◽  
Artinian Ring ◽  
Clique Number ◽  
Content Type ◽  
Cut Vertex ◽  
Vertex Set ◽  
Regular Character

PurposeThe authors study the interdisciplinary relation between graph and algebraic structure ring defining a new graph, namely “non-essential sum graph”. The nonessential sum graph, denoted by NES(R), of a commutative ring R with unity is an undirected graph whose vertex set is the collection of all nonessential ideals of R and any two vertices are adjacent if and only if their sum is also a nonessential ideal of R.Design/methodology/approachThe method is theoretical.FindingsThe authors obtain some properties of NES(R) related with connectedness, diameter, girth, completeness, cut vertex, r-partition and regular character. The clique number, independence number and domination number of NES(R) are also found.Originality/valueThe paper is original.

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