Publication Date: 2011
Match (03406253)65(1)pp. 27-32
Gutman and Zhou (Relations between Wiener, hyper-Wiener and Zagreb indices, Chemical Physics Letters 394 (2004) 93-95) obtained some bounds on Wiener and hyper-Wiener indices, in term of the first Zagreb index in molecular graphs with girth greater than four. We obtain new inequalities for Wiener and hyperWiener indices, in terms of the first and the second Zagreb indices and the number of hexagons in these graphs. These inequalities improve the bounds obtained by Gutman and Zhou and are the best possible bounds. Using these relations we obtain a bound on the second Zagreb index in terms of the first Zagreb index, for hexagon-free graphs.
Publication Date: 2012
Archivum Mathematicum (00448753)48(4)pp. 291-299
In this paper we study commutative rings R whose prime ideals are direct sums of cyclic modules. In the case R is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it is shown that for a local ring (R, Ai), the following statements are equivalent: (1) Every prime ideal of R is a direct sum of cyclic R-modules; (2) M =⊕λχλ Rωλ where λ is an index set and R/Ann(ωλ) is a principal ideal ring for each λ χ λ; (3) Every prime ideal of H is a direct sum of at most |A| cyclic fi-modules where A is an index set and M =⊕λχλ Rωλ; and (4) Every prime ideal of R is a summand of a direct sum of cyclic R-modules. Also, we establish a theorem which state that, to check whether every prime ideal in a Noetherian local ring (R, M) is a direct sum of (at most n) principal ideals, it suffices to test only the maximal ideal M.
Publication Date: 2012
Discrete Mathematics (0012365X)312(22)pp. 3349-3356
For a set W of vertices and a vertex v in a connected graph G, the k-vector rW(v)=(d(v,w1),⋯,d(v,wk)) is the metric representation of v with respect to W, where W=w1,⋯,wk and d(x,y) is the distance between the vertices x and y. The set W is a resolving set for G if distinct vertices of G have distinct metric representations with respect to W. The minimum cardinality of a resolving set for G is its metric dimension. In this paper, we study the metric dimension of the lexicographic product of graphs G and H, denoted by G[H]. First, we introduce a new parameter, the adjacency dimension, of a graph. Then we obtain the metric dimension of G[H] in terms of the order of G and the adjacency dimension of H. © 2012 Elsevier B.V. All rights reserved.
Publication Date: 2013
Communications in Algebra (00927872)41(12)pp. 4559-4575
This paper investigates the structure of cyclically pure (or C-pure) projective modules. In particular, it is shown that a ring R is left Noetherian if and only if every C-pure projective left R-module is pure projective. Also, over a left hereditary Noetherian ring R, a left R-module M is C-pure projective if and only if M = N ⊕ P, where N is a direct sum of cyclic modules and P is a projective left R-module. The relationship Cpurity with purity and RD-purity are also studied. It is shown that if R is a local duoring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is a principal ideal ring. If R is a left perfect duo-ring, then the C-pure projective left R-modules and the pure projective left R-modules coincide if and only if R is left Köthe (i.e., every left R-module is a direct sum of cyclic modules). Also, it is shown that for a ring R, if every C-pure projective left R-module is RD-projective, then R is left Noetherian, every p-injective left R-module is injective and every p-flat right R-module is flat. Finally, it is shown that if R is a left p.p-ring and every C-pure projective left R-module is RD-projective, then R is left Noetherian hereditary. The converse is also true when R is commutative, but it does not hold in the noncommutative case. © Taylor & Francis Group, LLC.
