scholarly journals Perfect Codes Over Induced Subgraphs of Unit Graphs of Ring of Integers Modulo n

2021 ◽  
Vol 20 ◽  
pp. 399-403
Author(s):  
Mohammad Hassan Mudaber ◽  
Nor Haniza Sarmin ◽  
Ibrahim Gambo
Keyword(s):  
Perfect Code ◽  
Perfect Codes ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Ring Of Integers ◽  
Unit Graph ◽  
Vertex Set

The induced subgraph of a unit graph with vertex set as the idempotent elements of a ring R is a graph which is obtained by deleting all non idempotent elements of R. Let C be a subset of the vertex set in a graph Γ. Then C is called a perfect code if for any x, y ∈ C the union of the closed neighbourhoods of x and y gives the the vertex set and the intersection of the closed neighbourhoods of x and y gives the empty set. In this paper, the perfect codes in induced subgraphs of the unit graphs associated with the ring of integer modulo n, Zn that has the vertex set as idempotent elements of Zn are determined. The rings of integer modulo n are classified according to their induced subgraphs of the unit graphs that accept a subset of a ring Zn of different sizes as the perfect codes

INDUCED SUBGRAPHS OF GAMMA GRAPHS

2013 ◽  
Vol 05 (03) ◽  
pp. 1350012 ◽  
Author(s):  
N. SRIDHARAN ◽  
S. AMUTHA ◽  
S. B. RAO
Keyword(s):  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Vertex Set ◽  
Isomorphic Graphs

Let G be a graph. The gamma graph of G denoted by γ ⋅ G is the graph with vertex set V(γ ⋅ G) as the set of all γ-sets of G and two vertices D and S of γ ⋅ G are adjacent if and only if |D ∩ S| = γ(G) – 1. A graph H is said to be a γ-graph if there exists a graph G such that γ ⋅ G is isomorphic to H. In this paper, we show that every induced subgraph of a γ-graph is also a γ-graph. Furthermore, if we prove that H is a γ-graph, then there exists a sequence {Gn} of non-isomorphic graphs such that H = γ ⋅ Gn for every n.

scholarly journals Perfect codes in power graphs of finite groups

2017 ◽  
Vol 15 (1) ◽  
pp. 1440-1449 ◽  
Author(s):  
Xuanlong Ma ◽  
Ruiqin Fu ◽  
Xuefei Lu ◽  
Mengxia Guo ◽  
Zhiqin Zhao
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Cyclic Subgroup ◽  
The Other ◽  
Perfect Code ◽  
Perfect Codes ◽  
Power Graph ◽  
Vertex Set ◽  
A Finite Group

Abstract The power graph of a finite group is the graph whose vertex set is the group, two distinct elements being adjacent if one is a power of the other. The enhanced power graph of a finite group is the graph whose vertex set consists of all elements of the group, in which two vertices are adjacent if they generate a cyclic subgroup. In this paper, we give a complete description of finite groups with enhanced power graphs admitting a perfect code. In addition, we describe all groups in the following two classes of finite groups: the class of groups with power graphs admitting a total perfect code, and the class of groups with enhanced power graphs admitting a total perfect code. Furthermore, we characterize several families of finite groups with power graphs admitting a perfect code, and several other families of finite groups with power graphs which do not admit perfect codes.

On the normalized Laplacian spectrum of the comaximal graphs

2021 ◽  
pp. 2250094
Author(s):  
Mojgan Afkhami
Keyword(s):  
Commutative Ring ◽  
Simple Graph ◽  
Laplacian Spectrum ◽  
Induced Subgraph ◽  
Randic Index ◽  
Ring Of Integers ◽  
Laplacian Energy ◽  
Comaximal Graph ◽  
Vertex Set ◽  
Nonzero Identity

Let [Formula: see text] be a commutative ring with nonzero identity. The comaximal graph of [Formula: see text], denoted by [Formula: see text], is a simple graph with 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]. Let [Formula: see text] be an induced subgraph of [Formula: see text] with nonunit elements of [Formula: see text] as vertices. In this paper, we describe the normalized Laplacian spectrum of [Formula: see text], and we determine it for some values of [Formula: see text], where [Formula: see text] is the ring of integers modulo [Formula: see text]. Moreover, we investigate the normalized Laplacian energy and general Randic index of [Formula: see text].

scholarly journals The stable subgroup graph

2018 ◽  
Vol 36 (3) ◽  
pp. 129-139
Author(s):  
Behnaz Tolue
Keyword(s):  
Abelian Group ◽  
Soluble Group ◽  
Normal Subgroups ◽  
Induced Subgraph ◽  
Vertex Set

In this paper we introduce stable subgroup graph associated to the group $G$. It is a graph with vertex set all subgroups of $G$ and two distinct subgroups $H_1$ and $H_2$ are adjacent if $St_{G}(H_1)\cap H_2\neq 1$ or $St_{G}(H_2)\cap H_1\neq 1$. Its planarity is discussed whenever $G$ is an abelian group, $p$-group, nilpotent, supersoluble or soluble group. Finally, the induced subgraph of stable subgroup graph with vertex set whole non-normal subgroups is considered and its planarity is verified for some certain groups.

scholarly journals Matrix Partitions with Finitely Many Obstructions

10.37236/976 ◽  
2007 ◽  
Vol 14 (1) ◽  
Author(s):  
Tomás Feder ◽  
Pavol Hell ◽  
Wing Xie
Keyword(s):  
Symmetric Matrix ◽  
Forbidden Subgraph ◽  
Input Graph ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
Forbidden Induced Subgraphs ◽  
Extra Effort ◽  
Matrix Partition ◽  
Finite Set ◽  
Forbidden Induced Subgraph

Each $m$ by $m$ symmetric matrix $M$ over $0, 1, *$, defines a partition problem, in which an input graph $G$ is to be partitioned into $m$ parts with adjacencies governed by $M$, in the sense that two distinct vertices in (possibly equal) parts $i$ and $j$ are adjacent if $M(i,j)=1$, and nonadjacent if $M(i,j)=0$. (The entry $*$ implies no restriction.) We ask which matrix partition problems admit a characterization by a finite set of forbidden induced subgraphs. We prove that matrices containing a certain two by two diagonal submatrix $S$ never have such characterizations. We then develop a recursive technique that allows us (with some extra effort) to verify that matrices without $S$ of size five or less always have a finite forbidden induced subgraph characterization. However, we exhibit a six by six matrix without $S$ which cannot be characterized by finitely many induced subgraphs. We also explore the connection between finite forbidden subgraph characterizations and related questions on the descriptive and computational complexity of matrix partition problems.

Algorithmic Aspects of Outer-Independent Total Roman Domination in Graphs

2021 ◽  
pp. 1-9
Author(s):  
Amit Sharma ◽  
P. Venkata Subba Reddy
Keyword(s):  
Linear Time ◽  
Domination Number ◽  
Chordal Graphs ◽  
Roman Domination ◽  
Induced Subgraph ◽  
Bounded Treewidth ◽  
Roman Domination Number ◽  
Vertex Set ◽  
Roman Dominating Function ◽  
Domination In Graphs

For a simple, undirected graph [Formula: see text], a function [Formula: see text] which satisfies the following conditions is called an outer-independent total Roman dominating function (OITRDF) of [Formula: see text] with weight [Formula: see text]. (C1) For all [Formula: see text] with [Formula: see text] there exists a vertex [Formula: see text] such that [Formula: see text] and [Formula: see text], (C2) The induced subgraph with vertex set [Formula: see text] has no isolated vertices and (C3) The induced subgraph with vertex set [Formula: see text] is independent. For a graph [Formula: see text], the smallest possible weight of an OITRDF of [Formula: see text] which is denoted by [Formula: see text], is known as the outer-independent total Roman domination number of [Formula: see text]. The problem of determining [Formula: see text] of a graph [Formula: see text] is called minimum outer-independent total Roman domination problem (MOITRDP). In this article, we show that the problem of deciding if [Formula: see text] has an OITRDF of weight at most [Formula: see text] for bipartite graphs and split graphs, a subclass of chordal graphs is NP-complete. We also show that MOITRDP is linear time solvable for connected threshold graphs and bounded treewidth graphs. Finally, we show that the domination and outer-independent total Roman domination problems are not equivalent in computational complexity aspects.

Finite commutative rings with higher genus unit graphs

2014 ◽  
Vol 14 (01) ◽  
pp. 1550002 ◽  
Author(s):  
Huadong Su ◽  
Kenta Noguchi ◽  
Yiqiang Zhou
Keyword(s):  
Nonnegative Integer ◽  
Commutative Rings ◽  
Orientable Surface ◽  
Simple Graph ◽  
Isomorphism Classes ◽  
Unit Graph ◽  
Higher Genus ◽  
Vertex Set ◽  
Finite Commutative Rings

Let R be a ring with identity. The unit graph of R, denoted by G(R), is a simple graph with vertex set R, and where two distinct vertices x and y are adjacent if and only if x + y is a unit in R. The genus of a simple graph G is the smallest nonnegative integer g such that G can be embedded into an orientable surface Sg. In this paper, we determine all isomorphism classes of finite commutative rings whose unit graphs have genus at most three.

scholarly journals Large Induced Subgraphs with All Degrees Odd

1992 ◽  
Vol 1 (4) ◽  
pp. 335-349 ◽  
Author(s):  
A. D. Scott
Keyword(s):  
Chromatic Number ◽  
Random Graphs ◽  
Connected Graph ◽  
Analogous Problem ◽  
Induced Subgraphs ◽  
Induced Subgraph

We prove that every connected graph of order n ≥ 2 has an induced subgraph with all degrees odd of order at least cn/log n, where cis a constant. We also give a bound in terms of chromatic number, and resolve the analogous problem for random graphs.

scholarly journals Graphs and complete intersection toric ideals

2015 ◽  
Vol 14 (09) ◽  
pp. 1540011 ◽  
Author(s):  
I. Bermejo ◽  
I. García-Marco ◽  
E. Reyes
Keyword(s):  
Complete Intersection ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Point Of View ◽  
Toric Ideals ◽  
Induced Subgraphs ◽  
Toric Ideal ◽  
Vertex Set ◽  
Combinatorial Point ◽  
Ring Graph

Our purpose is to study the family of simple undirected graphs whose toric ideal is a complete intersection from both an algorithmic and a combinatorial point of view. We obtain a polynomial time algorithm that, given a graph G, checks whether its toric ideal PG is a complete intersection or not. Whenever PG is a complete intersection, the algorithm also returns a minimal set of generators of PG. Moreover, we prove that if G is a connected graph and PG is a complete intersection, then there exist two induced subgraphs R and C of G such that the vertex set V(G) of G is the disjoint union of V(R) and V(C), where R is a bipartite ring graph and C is either the empty graph, an odd primitive cycle, or consists of two odd primitive cycles properly connected. Finally, if R is 2-connected and C is connected, we list the families of graphs whose toric ideals are complete intersection.

scholarly journals Long cycles in hypercubes with distant faulty vertices

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Petr Gregor ◽  
Riste Škrekovski
Keyword(s):  
Linear Code ◽  
Minimum Distance ◽  
Hamiltonian Cycle ◽  
Minimum Degree ◽  
Induced Subgraphs ◽  
Induced Subgraph ◽  
International Audience ◽  
Long Cycles

Graphs and Algorithms International audience In this paper, we study long cycles in induced subgraphs of hypercubes obtained by removing a given set of faulty vertices such that every two faults are distant. First, we show that every induced subgraph of Q(n) with minimum degree n - 1 contains a cycle of length at least 2(n) - 2(f) where f is the number of removed vertices. This length is the best possible when all removed vertices are from the same bipartite class of Q(n). Next, we prove that every induced subgraph of Q(n) obtained by removing vertices of some given set M of edges of Q(n) contains a Hamiltonian cycle if every two edges of M are at distance at least 3. The last result shows that the shell of every linear code with odd minimum distance at least 3 contains a Hamiltonian cycle. In all these results we obtain significantly more tolerable faulty vertices than in the previously known results. We also conjecture that every induced subgraph of Q(n) obtained by removing a balanced set of vertices with minimum distance at least 3 contains a Hamiltonian cycle.

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