scholarly journals On Symmetry of Complete Graphs over Quadratic and Cubic Residues

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
M. Haris Mateen ◽  
M. Khalid Mahmood ◽  
Shahbaz Ali ◽  
M. D. Ashraful Alam

In this study, we investigate two graphs, one of which has units of a ring Z n as vertices (or nodes) and an edge will be built between two vertices u and v if and only if u 3 ≡ v 3 mod   n . This graph will be termed as cubic residue graph. While the other is called Gaussian quadratic residue graph whose vertices are the elements of a Gaussian ring Z n i of the form α = a + i b , β = c + i    d , where a , b , c , d are the units of Z n . Two vertices α and β are adjacent to each other if and only if α 2 ≡ β 2 mod   n . In this piece of work, we characterize cubic and Gaussian quadratic residue graphs for each positive integer n in terms of complete graphs.

10.37236/4673 ◽  
2015 ◽  
Vol 22 (1) ◽  
Author(s):  
Alan Frieze ◽  
Wesley Pegden

We consider the question of the existence of homomorphisms between $G_{n,p}$ and odd cycles when $p=c/n$, $1<c\leq 4$. We show that for any positive integer $\ell$, there exists $\epsilon=\epsilon(\ell)$ such that if $c=1+\epsilon$ then w.h.p. $G_{n,p}$ has a homomorphism from $G_{n,p}$ to $C_{2\ell+1}$ so long as its odd-girth is at least $2\ell+1$. On the other hand, we show that if $c=4$ then w.h.p. there is no homomorphism from $G_{n,p}$ to $C_5$. Note that in our range of interest, $\chi(G_{n,p})=3$ w.h.p., implying that there is a homomorphism from $G_{n,p}$ to $C_3$.  These results imply the existence of random graphs with circular chromatic numbers $\chi_c$ satisfying $2<\chi_c(G)<2+\delta$ for arbitrarily small $\delta$, and also that $2.5\leq \chi_c(G_{n,\frac 4 n})<3$ w.h.p.


2019 ◽  
Vol 29 (04) ◽  
pp. 1950016
Author(s):  
Ajay Arora ◽  
Eddie Cheng ◽  
Colton Magnant

An path that is edge-colored is called proper if no two consecutive edges receive the same color. A general graph that is edge-colored is called properly connected if, for every pair of vertices in the graph, there exists a properly colored path from one to the other. Given two vertices u and v in a properly connected graph G, the proper distance is the length of the shortest properly colored path from u to v. By considering a specific class of colorings that are properly connected for Cartesian products of complete and cyclic graphs, we present results on the proper distance between all pairs of vertices in the graph.


1878 ◽  
Vol 27 (185-189) ◽  
pp. 381-383

In the “Proceedings of the Royal Society,” vol. xxvii, pp. 63-71, Professor Adams has given, by an inductive process, the development of the product of any two of Legendre’s Coefficients in a series of the Coefficients; from this is immediately deduced the value of the integral between the limits -1 and +1 of the product of any three of the Coefficients. On the other hand, if we know the value of this definite integral, we can immediately deduce the development of the product of any two of the Coefficients. Thus it may be of interest to give a brief investigation of the value of the definite integral. I follow the notation adopted by Professor Adams. The formula to be established is ∫ 1 -1 P m P n P p dμ = 2/2 s + 1 A ( s - m ) A ( s - n ) A ( s - p ) / A( s ), where 2 s = m + n + P , and the functional symbol A( r ) is thus defined: if r is a positive integer A ( r ) = 1. 3. 5. . (2 r - 1) /1. 2. 3. . r , and in all other cases A ( r ) is to be considered zero, except when r = 0, and then it is to be considered = 1.


2011 ◽  
Vol 10 (01) ◽  
pp. 73-99 ◽  
Author(s):  
PATRICK F. SMITH

Let R be a ring with identity. A unital left R-module M has the min-property provided the simple submodules of M are independent. On the other hand a left R-module M has the complete max-property provided the maximal submodules of M are completely coindependent, in other words every maximal submodule of M does not contain the intersection of the other maximal submodules of M. A semisimple module X has the min-property if and only if X does not contain distinct isomorphic simple submodules and this occurs if and only if X has the complete max-property. A left R-module M has the max-property if [Formula: see text] for every positive integer n and distinct maximal submodules L, Li (1 ≤ i ≤ n) of M. It is proved that a left R-module M has the complete max-property if and only if M has the max-property and every maximal submodule of M/Rad M is a direct summand, where Rad M denotes the radical of M, and in this case every maximal submodule of M is fully invariant. Various characterizations are given for when a module M has the max-property and when M has the complete max-property.


2016 ◽  
Vol 19 (6) ◽  
Author(s):  
Robert T. Curtis

AbstractThe large Conway simple groupRemarkably, we can start at the other end in the sense that if we considerSpecifically, we associate with each directed edgeThus this is not simply a sequence of nested subgroups in a larger group, but a finite family of closely-related perfect groups.


1962 ◽  
Vol 27 (2) ◽  
pp. 195-211 ◽  
Author(s):  
Richard Montague

The present paper concerns the relation of relative interpretability introduced in [8], and arises from a question posed by Tarski: are there two finitely axiomatizable subtheories of the arithmetic of natural numbers neither of which is relatively interpretable in the other? The question was answered affirmatively (without proof) in [3], and the answer was generalized in [4]: for any positive integer n, there exist n finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. A further generalization was announced in [5] and is proved here: there is an infinite set of finitely axiomatizable subtheories of arithmetic such that no one of them is relatively interpretable in the union of the remainder. Several lemmas concerning the existence of self-referential and mutually referential formulas are given in Section 1, and will perhaps be of interest on their own account.


2011 ◽  
Vol Vol. 13 no. 2 (Graph and Algorithms) ◽  
Author(s):  
Eva Jelinkova ◽  
Ondrej Suchy ◽  
Petr Hlineny ◽  
Jan Kratochvil

Graphs and Algorithms International audience Seidel's switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other by a sequence of switches. In this paper, we continue the study of computational complexity aspects of Seidel's switching, concentrating on Fixed Parameter Complexity. Among other results we show that switching to a graph with at most k edges, to a graph of maximum degree at most k, to a k-regular graph, or to a graph with minimum degree at least k are fixed parameter tractable problems, where k is the parameter. On the other hand, switching to a graph that contains a given fixed graph as an induced subgraph is W [1]-complete. We also show the NP-completeness of switching to a graph with a clique of linear size, and of switching to a graph with small number of edges. A consequence of the latter result is the NP-completeness of Maximum Likelihood Decoding of graph theoretic codes based on complete graphs.


10.37236/1994 ◽  
2012 ◽  
Vol 19 (1) ◽  
Author(s):  
Rani Hod ◽  
Marcin Krzywkowski

A team of $n$ players plays the following game. After a strategy session, each player is randomly fitted with a blue or red hat. Then, without further communication, everybody can try to guess simultaneously his own hat color by looking at the hat colors of the other players. Visibility is defined by a directed graph; that is, vertices correspond to players, and a player can see each player to whom he is connected by an arc. The team wins if at least one player guesses his hat color correctly, and no one guesses his hat color wrong; otherwise the team loses. The team aims to maximize the probability of a win, and this maximum is called the hat number of the graph.Previous works focused on the hat problem on complete graphs and on undirected graphs. Some cases were solved, e.g., complete graphs of certain orders, trees, cycles, and bipartite graphs. These led Uriel Feige to conjecture that the hat number of any graph is equal to the hat number of its maximum clique.We show that the conjecture does not hold for directed graphs. Moreover, for every value of the maximum clique size, we provide a tight characterization of the range of possible values of the hat number. We construct families of directed graphs with a fixed clique number the hat number of which is asymptotically optimal. We also determine the hat number of tournaments to be one half.


1999 ◽  
Vol 29 (2) ◽  
pp. 311-314 ◽  
Author(s):  
Bjørn Sundt

In the present discussion we point out the relation of some results in Dickson & Waters (1999) to similar results in Sundt (1998a, b).We shall need some notation. For a positive integer m let m be the set of all m × 1 vectors with positive integer-valued elements and m+ = m ~ {0}. A vector will be denoted by a bold-face letter and each of its elements by the corresponding italic with a subscript denoting the number of the elements; the subscript · denotes the sum of the elements. Let m0 be the class of probability functions on m with a positive probability at 0 and m+ the class of probability functions on m+. For j = 1,…, m we introduce the m × 1 vector ej where the jth element is one and all the other elements zero. We make the convention that summation over an empty range is equal to zero.Let g ∈ m0 be the compound probability function with counting distribution with probability function v ∈ 10 and severity distribution with probability function h ∈ m+; we shall denote this compound probability function by v V h. Sundt (1998a) showed thatwhere φv denotes the De Pril transform of v, given byMotivated by (2) Sundt (1998a) defined the De Pril transform φg of g byThis defines the De Pril transform for all probability functions in m0. Insertion of (2) in (3) givesand by solving φg(X) we obtainSundt (1998a) studies the De Pril transform defined in this way and found in particular that it is additive for convolutions.


1995 ◽  
Vol 37 (1) ◽  
pp. 69-71 ◽  
Author(s):  
Howard Smith

Given a group G and a positive integer k, let vk(G) denote the number of conjugacy classes of subgroups of G which are not subnormal of defect at most k. Groups G such that vkG) < ∝ for some k are considered in Section 2 of [1], and Theorem 2.4 of that paper states that an infinite group G for which vk(G) < ∝ (for some k) is nilpotent provided only that all chief factors of G are locally (soluble or finite). Now it is easy to see that a group G whose chief factors are of this type is locally graded, that is, every nontrivial, finitely generated subgroup F of G has a nontrivial finite image (since there is a chief factor H/K of G such that F is contained in H but not in K). On the other hand, every (locally) free group is locally graded and so there is in general no restriction on the chief factors of such groups. The class of locally graded groups is a suitable class to consider if one wishes to do no more than exclude the occurrence of finitely generated, infinite simple groups and, in particular, Tarski p-groups. As pointed out in [1], Ivanov and Ol'shanskiĭ have constructed (finitely generated) infinite simple groups all of whose proper nontrivial subgroups are conjugate; clearly a group G with this property satisfies v1(G) = l. The purpose of this note is to provide the following generalization of the above-mentioned theorem from [1].


Sign in / Sign up

Export Citation Format

Share Document