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Articles
Applied Mathematics Letters (18735452)22(10)pp. 1571-1576
In this work we show that among all n-vertex graphs with edge or vertex connectivity k, the graph G = Kk ∨ (K1 + Kn - k - 1), the join of Kk, the complete graph on k vertices, with the disjoint union of K1 and Kn - k - 1, is the unique graph with maximum sum of squares of vertex degrees. This graph is also the unique n-vertex graph with edge or vertex connectivity k whose hyper-Wiener index is minimum. © 2009 Elsevier Ltd. All rights reserved.
Discrete Applied Mathematics (0166218X)158(3)pp. 219-221
A block graph is a graph whose blocks are cliques. For each edge e = u v of a graph G, let Ne (u) denote the set of all vertices in G which are closer to u than v. In this paper we prove that a graph G is a block graph if and only if it satisfies two conditions: (a) The shortest path between any two vertices of G is unique; and (b) For each edge e = u v ∈ E (G), if x ∈ Ne (u) and y ∈ Ne (v), then, and only then, the shortest path between x and y contains the edge e. This confirms a conjecture of Dobrynin and Gutman [A.A. Dobrynin, I. Gutman, On a graph invariant related to the sum of all distances in a graph, Publ. Inst. Math., Beograd. 56 (1994) 18-22]. © 2009 Elsevier B.V. All rights reserved.
Bulletin of the Korean Mathematical Society (10158634)(4)pp. 663-673
This work was intended as an attempt to introduce and investigate the Connes-amenability of module extension of dual Banach algebras. It is natural to try to study the weak*-continuous derivations on the module extension of dual Banach algebras and also the weak Connesamenability of such Banach algebras. © 2010 The Korean Mathematical Society.
Journal of Algebra (00218693)345(1)pp. 257-265
A theorem from commutative algebra due to Köthe and Cohen-Kaplansky states that, "a commutative ring R has the property that every R-module is a direct sum of cyclic modules if and only if R is an Artinian principal ideal ring". Therefore, an interesting natural question of this sort is "whether the same is true if one only assumes that every ideal is a direct sum of cyclic modules?" The goal of this paper is to answer this question in the case R is a finite direct product of commutative Noetherian local rings. The structure of such rings is completely described. In particular, this yields characterizations of all commutative Artinian rings with this property. © 2011.
Match (03406253)65(1)pp. 71-78
The Wiener index of a simple graph is defined as the sum of distances between all vertices of the graph. It is well known that the Wiener index of a tree can be obtained as an edge additive quantity where edge contributions are given as the product of the number of vertices closer to each of the two end points of each edge. Thus the distances between vertices are not used for computing the Wiener index of trees. In a similar manner we introduce new topological indices which yields the Wiener, hyper-Wiener, Schultz and modified Schultz indices as special cases for trees. One advantage of this method is that in computing Schultz and modified Schultz of trees we need not take in to account the distances between vertices.
Applied Mathematics Letters (18735452)24(10)pp. 1625-1629
For an ordered set W=w1,w2,⋯,wk of vertices and a vertex v in a connected graph G, the ordered k-vector r(v|W):=(d(v,w1),d(v,w2),⋯,d(v,wk)) is called the (metric) representation of v with respect to W, where d(x,y) is the distance between the vertices x and y. The set W is called a resolving set for G if distinct vertices of G have distinct representations with respect to W. A resolving set for G with minimum cardinality is called a basis of G and its cardinality is the metric dimension of G. A connected graph G is called a randomly k-dimensional graph if each k-set of vertices of G is a basis of G. In this work, we study randomly k-dimensional graphs and provide some properties of these graphs. © 2011 Elsevier Ltd. All rights reserved.
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.
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.
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.
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.
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