In computer science, graphs are used in variety of applications directly or indirectly. Especially quantitative labeled graphs have played a vital role in computational linguistics, decision making software tools, coding theory and path determination in networks. For a graph G(V, E) with the vertex set V and the edge set E, a vertex k-labeling φ : V → {1, 2, . . . , k} is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f their w φ (e) = w φ (f ), where the weight of an edge e = xy ∈ E(G) is w φ (xy) = φ(x) + φ(y). The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we determine the edge irregularity strengths of some chain graphs and the join of two graphs. We introduce a conjecture and open problems for researchers for further research.
In this paper we study M -small principally injective (in short, M -sp-injective) module which is the generalization of M -principally injective module. We prove that if M is finite dimensional and quasi-sp-injective then its endomorphism ring S is semi-local ring. We characterize the M -sp-injective module with the help of epi-retractable modules. Keywords: Small submodule; M -cyclic submodule; M -sp-injective modules; quasi-spinjective modules and epi-retractable module. AMS Subject Classification: 16D10, 16D50, 16D60 1250005-1 V. Kumar et al.N . A module M is called quasi-principally (or semi) injective, if it is M -principally injective. In this paper we introduce the notion of M -small principally injective modules and quasi-small principally injective modules which we abbreviate as Msp-injective and quasi-sp-injective modules. A submodule K of an R-module M is said to be small in M , written as K M , if for every submodule L ⊆ M withFor details see [1,5]. In this paper, we study M -sp-injective modules and M -sp-injective rings and also give an example of an M -sp-injective module which is not an M -principally injective module.Consider the following condition for an R-module M ., it is called continuous, if it satisfies (C 1 ) and (C 2 ), and quasi-continuous, if it satisfies (C 1 ) and (C 3 ). For undefined notations and terminologies see [1,14]. M-Small Principally Injective ModulesGeneralizing the notion of Sanh et al. [13], we now give a new definition. DefinitionAn R-module N is called M -small principally injective, if for every small M -cyclic submodule K of M , any homomorphism from K to N can be extended to a homomorphism from M to N . If M is a M -small principally injective module, then it is called a quasi-small principally injective module, or M -sp-injective module, and for short a ring R is called a right small-principally injective ring, if R R as a right R-module. Example(1) Every M -principally injective module is a M -sp-injective module.(2) Every semi-simple module is an M -sp-injective module.We now give an example of M -sp-injective modules which is not M -principally injective. ExampleZ is a Z-small principally injective module, but it is not Z-principally injective, because the only small Z-cyclic submodule of Z is 0.The following lemmas are the generalizations of [13, Lemmas 2.2-2.4]. Lemma 2.1. Let M i (1 ≤ i ≤ n) be M-sp-injective modules. Then n i=1 M i is M -sp-injective module. 1250005-2 M -SP-Injective Modules Proof. The proof is similar to that of [13, Lemma 2.2]. Lemma 2.2. Let X be an M-cyclic submodule of M . If X is M-sp-injective module, then it is a direct summand of M . Proof. The proof is similar to that of [13, Lemma 2.3]. Lemma 2.3. Every direct summand of M-sp-injective module is an M-sp-injective module. Proof. By the same argument as that given in [13, Lemma 2.4]. 1250005-5 V. Kumar et al. Proposition 2.6. For a hollow R-module M, the following conditions are equivalent :(1) M is quasi-sp-injective module;(2) M is quasi-principally injective module.Proof. It is ...
In this paper, the concept of quasi-pseudo principally injective modules is introduced and a characterization of commutative semi-simple rings is given in terms of quasi-pseudo principally injective modules. An example of pseudo M -p-injective module which is not M -pseudo injective is given.
A graph G is called edge-magic if there exists a bijective function φ : V (G) ∪ E(G) → {1, 2,. .. , |V (G)| + |E(G)|} such that φ(x) + φ(xy) + φ(y) = c(φ) is a constant for every edge xy ∈ E(G), called the valence of φ. Moreover, G is said to be super edge-magic if φ(V (G)) = {1, 2,. .. , |V (G)|}. The super edge-magic deciency of a graph G, denoted by µs(G), is the minimum nonnegative integer n such that G ∪ nK1, has a super edge-magic labeling, if such integer does not exist we dene µs(G) to be +∞. In this paper, we study the super edge-magic deciency of some Toeplitz graphs.
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