Juan Rada

Energy of digraphs

We extend the concept of energy to directed graphs in such a way that Coulsons integral formula remains valid. As a consequence, it is shown that the energy is increasing over the set of digraphs with n vertices and cycles of length h, with respect to a quasi-order relation. Applications to the problem of extremal values of the energy in various classes of digraphs are considered.

Polygonal chains with minimal energy

Abstract The energy of a graph G is defined as E ( G )=∑ i =1 p | λ i |, where λ i ( i =1,…, p ) are the eigenvalues of the adjacency matrix of G . We show that among all polygonal chains with polygons of 4 n −2 vertices ( n ⩾2), the linear polygonal chain has minimal energy.

Energy ordering of catacondensed hexagonal systems

The energy of a graph G is defined as E(G) = Σi=1n |λi|, where λi (i = 1,..., n) are the eigenvalues of G. In this work we define the coalescence of two graphs with respect to (oriented) edges, and show that for the graphs X and Y in Fig. 2, which are obtained by coalescence of bipartite graphs around the six-vertex cycle C6, E(X)≥E(Y). As a by-product, we give energy ordering relations in the class of catacondensed hexagonal systems.

Cotorsion Pairs in C(R-Mod)

In [8] Salce introduced the notion of a cotorsion pair (A,B) in the category of abelian groups. But his definitions and basic results carry over to more general abelian categories and have proved useful in a variety of settings. In this article we will consider complete cotorsion pairs (C,D) in the category C(R-Mod) of complexes of left R-modules over some ring R. If (C,D) is such a pair, and if C is closed under taking suspensions, we will show when we regard K(C) and K(D) as subcategories of the homotopy category K(RMod), then the embedding functors K(C) → K(R-Mod) and K(D) → K(R-Mod) have left and right adjoints, respectively. In finding examples of such pairs, we will describe a procedure for using Hoveys results in [5] to find a new model structure on C(R-Mod).

Higher order connectivity index of starlike trees

We show that for every integer h ≥ 0, the higher order connectivity index hχ(T) of a starlike tree T (a tree with unique vertex of degree > 2) is completely determined by its branches of length ≤ h. As a consequence, we show that starlike trees which have equal h-connectivity index for all h ≥ 0 are isomorphic.

Wiener index of Eulerian graphs

The Wiener index of a connected graph G is the sum of distances between all pairs of vertices of G . We characterize Eulerian graphs (with a fixed number of vertices) with smallest and greatest Wiener indices.

Matchings in starlike trees

Abstract Let m ( G,k ) be the number of k -matchings in the graph G . We write G 1 ⪯ G 2 if m ( G 1 , k ) ≤ m ( G 2 , k ) for all k = 1, 2,…. A tree is said to be starlike if it possesses exactly one vertex of degree greater than two. The relation T 1 ⪯ T 2 is shown to hold for various pairs of starlike trees T 1 , T 2 . The starlike trees (with a given number of vertices), extremal with respect to the relation ⪯, are characterized.

Variation of the Wiener index under tree transformations

The Wiener index W(T) is defined as the sum of distances between all pairs of vertices of the tree T. In this paper we find the variation of the Wiener index under certain tree transformations, which can be described in terms of coalescence of trees. As a consequence, conditions for nonisomorphic trees having equal Wiener index are presented. Also, a partial order on the collection of trees (with a fixed number of vertices) is introduced, providing structural information about the behavior of W.

On semiregular rings whose finitely generated modules embed in free modules

We consider rings as in the title and find the precise obstacle for them not to be Quasi-Frobenius, thus shedding new light on an old open question in Ring Theory. We also find several partial affirmative answers for that question. This paper was finished while Juan Rada was preparing his Ph.D. at the Universidad de Murcia. Manuel Saorı́n was partially supported by D.G.I.C.Y.T. (PB93-0515, Spain) and the Comunidad Autónoma de Murcia (PIB 94-25). Received by the editors October 24, 1995. AMS subject classification: Primary: 16D10, 16L60; Secondary: 16N20. c Canadian Mathematical Society 1997.

Randić index and lexicographic order

AbstractLet T be a tree and consider the Randić index χ(T)=∑