On the inverse of the adjacency matrix of a graph

Farrugia, Alexander; Gauci, John Baptist; Sciriha, Irene

doi:10.2478/spma-2013-0006

Cited by 5 publications

(10 citation statements)

References 4 publications

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“…Thus, if ipso omni-insulators with odd n exist, they are non-bipartite (odd bipartite graphs have odd η ≥ 1) and must have at least two disjoint odd cycles, since deletion of a vertex leaves a graph with even order but η = 1, implying a non-bipartite graph. Furthermore, a construction for reducing ipso omni-insulators 38 (Algorithm 36 in that paper) implies that the such smallest graph has no pendant edge.…”

Section: Insulatorsmentioning

confidence: 99%

Omni-conducting and omni-insulating molecules

Fowler

Pickup

Todorova

et al. 2014

The Journal of Chemical Physics

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The source and sink potential model is used to predict the existence of omni-conductors (and omniinsulators): molecular conjugated π systems that respectively support ballistic conduction or show insulation at the Fermi level, irrespective of the centres chosen as connections. Distinct, ipso, and strong omni-conductors/omni-insulators show Fermi-level conduction/insulation for all distinct pairs of connections, for all connections via a single centre, and for both, respectively. The class of conduction behaviour depends critically on the number of non-bonding orbitals (NBO) of the molecular system (corresponding to the nullity of the graph). Distinct omni-conductors have at most one NBO; distinct omni-insulators have at least two NBO; strong omni-insulators do not exist for any number of NBO. Distinct omni-conductors with a single NBO are all also strong and correspond exactly to the class of graphs known as nut graphs. Families of conjugated hydrocarbons corresponding to chemical graphs with predicted omni-conducting/insulating behaviour are identified. For example, most fullerenes are predicted to be strong omni-conductors. © 2014 AIP Publishing LLC.

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Section: Insulatorsmentioning

confidence: 99%

Omni-conducting and omni-insulating molecules

Fowler

Pickup

Todorova

et al. 2014

The Journal of Chemical Physics

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“…The nullity of two-vertex-deleted subgraphs of NSSDs may be one of two values, as shown in Theorem 3.3 below. This result was proved in [1] for NSSDs which are not necessarily simple. Thus, if G is a NSSD, then for i = j, the ij th entry of A −1 is zero if G−i−j is singular and is nonzero if G − i − j is nonsingular.…”

Section: Nullity Ofmentioning

confidence: 70%

“…The purpose of this paper is to prove Conjecture 1.2 for the subclass of chemical graphs, whose vertex degree is at most three. In order to do this, we use the class of NSSDs (Non-Singular graphs with a Singular Deck) defined in [1], which is a superclass of that of nuciferous graphs. Definition 1.3.…”

Section: Introductionmentioning

confidence: 99%

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No chemical graph on more than two vertices is nuciferous

Sciriha¹,

Farrugia²

2016

Ars Math. Contemp.

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A simple graph is nuciferous if its 0-1 adjacency matrix is nonsingular and if its inverse has zero entries on its main diagonal and a non-zero entry at each off-diagonal position. A nuciferous graph is a molecular graph that represents an ipso omni-insulating but distinct omni-conducting molecule. It has been conjectured in 2012 that only K 2 , the complete graph on two vertices, is nuciferous. We show that this conjecture is true for chemical graphs, that is, graphs whose vertex degree is at most three. (Irene Sciriha), alex.farrugia@um.edu.mt (Alexander Farrugia) c b This work is licensed under http://creativecommons.org/licenses/by/3.0/

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“…, n are singular [5][6][7]. The term NSSD was introduced in [8], motivated by the search for carbon molecules in the Huckel model. The first step in the history of the development of hyperstructure theory was the 8th congress of Scandinavian mathematician from 1934, when Marty [9] put forward the concept of hypergroup, analyzed its properties and showed its utility in the study of groups, algebraic functions, and rational fractions.…”

Section: Introductionmentioning

confidence: 99%

A novel method to construct NSSD molecular graphs

et al. 2019

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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 H v -group. An algorithm is presented to construct the NSSD graphs from these commuting graphs.

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On the inverse of the adjacency matrix of a graph

Cited by 5 publications

References 4 publications

Omni-conducting and omni-insulating molecules

Omni-conducting and omni-insulating molecules

No chemical graph on more than two vertices is nuciferous

A novel method to construct NSSD molecular graphs

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