Constructing NSSD molecular graphs

A graph is said to be non-singular if it has no eigenvalue equal to zero; otherwise it is singular. Molecular graphs that are non-singular and also have the property that all subgraphs of them obtained by deleting a single vertex are themselves singular, known as NSSD graphs, are of importance in the theory of molecular π-electron conductors; NSSD = non-singular graph with a singular deck. Whereas all non-singular bipartite graphs (therefore, the molecular graphs of all closed-shell alternant conjugated hydrocarbons) are NSSD, the non-bipartite case is much more complicated. Only a limited number of non-bipartite molecular graphs have the NSSD property. Several methods for constructing such molecular graphs are presented.peer-reviewe

A novel method to construct NSSD molecular graphs

A graph is said to be NSSD (=non-singular with a singular deck) if it has no eigenvalue equal to zero, whereas all its vertex-deleted subgraphs have eigenvalues equal to zero. NSSD graphs are of importance in the theory of conductance of organic compounds. In this paper, a novel method is described for constructing NSSD molecular graphs from the commuting graphs of the Hv-group. An algorithm is presented to construct the NSSD graphs from these commuting graphsThis research is partially funded through Quaid-i-Azam University grant URF-201

The adjacency matrices of complete and nutful graphs

A real symmetric matrix G with zero entries on its diagonal is an adjacency matrix associated with a graph G (with weighted edges and no loops) if and only if the non-zero entries correspond to edges of G. An adjacency matrix G belongs to a generalized-nut graph G if every entry in a vector in the nullspace of G is non-zero. A graph G is termed NSSD if it corresponds to a non-singular adjacency matrix G with a singular deck {G- v}, where G- v is the submatrix obtained from G by deleting the vth row and column. An NSSD G whose deck consists of generalized- nut graphs with respect to G is referred to as a G-nutful graph. We prove that a G-nutful NSSD is equivalent to having a NSSD with G-1 as the adjacency matrix of the complete graph. If the entries of G for a G-nutful graph are restricted to 0 or 1, then the graph is known as nuciferous, a concept that has arisen in the context of the quantum mechanical theory of the conductivity of non-singular Carbon molecules according to the SSP model. We characterize nuciferous graphs by their inverse and the nullities of their one- and two-vertex deleted subgraphs. We show that a G-nutful graph is a NSSD which is either K2 or has no pendant edges. Moreover, we reconstruct a labelled NSSD either from the nullspace generators of the ordered one-vertex deleted subgraphs or from the determinants of the ordered two-vertex deleted subgraphs.peer-reviewe

Fixed-point free circle actions on 4-manifolds

This paper is concerned with fixed-point free $S^1$-actions (smooth or locally linear) on orientable 4-manifolds. We show that the fundamental group plays a predominant role in the equivariant classification of such 4-manifolds. In particular, it is shown that for any finitely presented group with infinite center, there are at most finitely many distinct smooth (resp. topological) 4-manifolds which support a fixed-point free smooth (resp. locally linear) $S^1$-action and realize the given group as the fundamental group. A similar statement holds for the number of equivalence classes of fixed-point free $S^1$-actions under some further conditions on the fundamental group. The connection between the classification of the $S^1$-manifolds and the fundamental group is given by a certain decomposition, called fiber-sum decomposition, of the $S^1$-manifolds. More concretely, each fiber-sum decomposition naturally gives rise to a Z-splitting of the fundamental group. There are two technical results in this paper which play a central role in our considerations. One states that the Z-splitting is a canonical JSJ decomposition of the fundamental group in the sense of Rips and Sela. Another asserts that if the fundamental group has infinite center, then the homotopy class of principal orbits of any fixed-point free $S^1$-action on the 4-manifold must be infinite, unless the 4-manifold is the mapping torus of a periodic diffeomorphism of some elliptic 3-manifold. The paper ends with two questions concerning the topological nature of the smooth classification and the Seiberg-Witten invariants of 4-manifolds admitting a smooth fixed-point free $S^1$-action.Comment: 42 pages, no figures, Algebraic and Geometric Topolog

Polynomial reconstruction : old and new techniques

The Polynomial Reconstruction Problem (PRP) asks whether for a graph G of order at least three, the characteristic polynomial can be reconstructed from the p-deck PD(G) of characteristic polynomials of the one-vertex-deleted subgraphs. The problem is still open in general but has been proved for certain classes of graphs. We discuss the tools and techniques most commonly used and survey the main positive results obtained so far, pointing out the classes of graphs for which we know that the PRP has a positive resolution.peer-reviewe

Effective distance between nested Margulis tubes

We give sharp, effective bounds on the distance between tori of fixed injectivity radius inside a Margulis tube in a hyperbolic 3-manifold.Comment: 25 pages, 3 figures. v3 contains minor revisions. To appear in Transactions of the AM

Nut graphs : maximally extending cores

A graph G is singular if there is a non-zero eigenvector v(0) in the nullspace of its adjacency matrix A. Then Av(0) = 0. The subgraph induced by the vertices corresponding to the non-zero components of v(0) is the core of G (w.r.t. v(0)). The set whose members are the remaining vertices of G is called the periphery(w.r.t. v(0)) and corresponds to the sere components of v(0). The dimension of the nullspace of A is called the nullity of G. This paper investigates nut graphs which are graphs of nullity one whose periphery is empty. It is shown that nut graphs of order n exist for each n greater than or equal to 7 and that among singular graphs nut graphs are characterized by their deck of spectra.peer-reviewe