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.


INTRODUCTION
Fowler et al. [11][12][13][14] developed a tight-binding model for describing the steady-state current created by the injection of ballistic electrons via external contacts into an unsaturated conjugated π-electron system, treated as a single-molecule conductor.Within this model, the conjugated molecule is said to be an ipso insulator if it behaves as an insulator for a particular connection via a single π-electron center.If this is the case with all conjugated centers, then the respective molecule is an ipso omni insulator; for details see Tables IV and IX, as well as Theorem 6.1 in the paper. [14]et G be the molecular graph of a conjugated πelectron system, [5,6] and let 1 2  , , , n λ λ λ  be the eigenvalues of its adjacency matrix (usually referred [5,6] to as the eigenvalues of the underlying graph).Recall that such graphs are connected and have vertices whose degrees do not exceed three. [5,6]he number of eigenvalues of the graph G that are equal to zero is called the nullity of G, and will be denoted by ( ) η G .A (molecular) graph whose nullity is greater than zero is said to be singular.
Denote by ( , ) G λ  the characteristic polynomial of the graph G.As is well known, [5,6] the graph G is singular if and only if ( ,0) 0 are the vertices of the graph G, then the collection of its vertex-deleted subgraphs, [17][18][19] If the graph G is non-singular, i.e., ( ) 0 η G  , but all elements of its deck are singular, i.e., ( ) 0 , 1,2, , G is "non-singular with a singular deck'', or, abbreviated, NSSD. [18]Because of the Cauchy interlacing theorem, [20] it follows that ( ) 1 , 1,2, , NSSD molecular graphs are interesting from a theoretical-chemistry point of view, because any such graph represents an ipso omni insulator, and vice-versa: the molecular graph of any conjugated π-electron system that is ipso omni insulator is NSSD.The proof of this result is found in Theorem 4.4 of Ref. 14 (necessity) and in Deductions 5.7 and 5.9 of the same paper.
In this paper, we examine the structure of NSSD molecular graphs and ways how these can be constructed.It is easy to show that the molecular graph of any alternant conjugated hydrocarbon without non-bonding molecular orbitals is an NSSD graph.Contrary to this, among molecular graphs of non-alternant conjugated hydrocarbons, a surprisingly small number has the NSSD property.Our main contribution is the construction of nonbipartite NDDS molecular graphs.

BIPARTITE NSSD MOLECULAR GRAPHS
At this point, we recall that a graph is said to be bipartite if it does not possess odd-membered cycles.Bipartite molecular graphs represent alternant conjugated π-electron systems.For such graphs, the classic Coulson-Rushbrooke pairing theorem holds, [5,6,21] according to which the eigenvalues are paired as 1 , 1,2, , tly, if the number of vertices n is odd, then at least one eigenvalue is equal to zero and the respective graph is singular.This has the following straightforward and elementary consequence: Proposition 1a.A bipartite graph is NSSD if and only if it is non-singular.There are no bipartite NSSD graphs with an odd number of vertices.
Within the Hückel (tight-binding) molecular orbital approximation, each zero eigenvalue of the molecular graph corresponds to a non-bonding molecular orbital (NBMO). [5,6]The absence of NBMOs is one of the necessary (but not sufficient) conditions for stability of the underlying conjugated π-electron system. [10]Thus, the chemical consequence of Proposition 1a is: Proposition 1b.A bipartite molecular graph is NSSD if it represents a stable alternant conjugated π-electron system with no NBMOs.Such graphs necessarily possess an even number of vertices.
According to Proposition 1b, each stable alternant conjugated hydrocarbon is associated with an NSSD graph and is therefore (within the Fowler model) an ipso omni insulator.Some characteristic examples are given in Figure 1.These coincide with the examples found in the paper by Fowler et al. [14]

NON-BIPARTITE NSSD MOLECULAR GRAPHS
Whereas the molecular graphs of all stable, closed-shell alternant hydrocarbons are NSSD, the situation with nonbipartite species is significantly different.Non-bipartite NSSD graphs happen to be relatively rare, and this especially applies to molecular graphs.
First of all, the data in Table 1 show that the vast majority of connected graphs are non-bipartite.
We have performed a systematic computer search for n-vertex non-bipartite NSSD graphs and molecular graphs, for all values of n up to n = 11.The data in Table 1 reveal that in the case n = 11, more than one billion graphs had to be tested.For this reason, such calculations for n > 11would be technically non-feasible.Results of similar kind, but restricted to a few classes of molecular graphs, were earlier reported by Fowler et al. [14] Our findings fully agree with these, and corroborate them.
Our findings for an odd number of vertices are the following: For n = 3,5,7,9, there are no non-bipartite NSSD graphs.For n = 11, there are exactly 15 non-bipartite NSSD graphs, depicted in Figure 2. None of these is a molecular graph.
Our findings for an even number of vertices are the following: For n = 4, there are no non-bipartite NSSD graphs.For n = 6, there exists a single non-bipartite NSSD molecular graph, the [3]radialene graph G1 in Figure 3.For n = 8, there exist exactly four non-bipartite NSSD molecular graphs (see Figure 3).For n = 10, out of over one million non-bipartite graphs, only some two dozen NSSD molecular graphs have been discovered.Their main representatives are depicted in Figure 4.       , , G G G do not exceed 3, but for chemical reasons these should not be considered as representing real conjugated πelectron systems.Proof.If u is a pendent vertex of a graph H, adjacent to the vertex v, then [8,10] Applying Eq. ( 1) repeatedly to all but one pendent vertices of G*, we end with the 2-vertex connected graph whose nullity is zero.Therefore, ( *) 0 η G  .If u is any pendent vertex of G*, then by applying Eq.( 1) repeatedly to all pendent vertices of G* -u, we end with the single-vertex graph whose nullity is one.Therefore, ( * ) 1 η G u   .If v is any non-pendent vertex of G*, then the subgraph * G v  contains an isolated vertex and is therefore singular.Thus, G*is non-singular, and all its vertex-deleted subgraphs are singular, i.e., G* is an NSSD graph.□ The only non-bipartite NSSD molecular graphs that can be obtained by means of Transformation 1 are the radialene graphs, see Figure 5.
Let GA be a graph and let be x one of its vertices.Let GB be another graph and y one of its vertices.Denote by GA(x,y)GB the graph obtained from GA and GB by joining the vertices x and y by a new edge.Transformation 2. If GA and GB are two NSSD graphs, and x and y any two of their vertices, then GA(x,y)GB is also an NSSD graph.Proof.The characteristic polynomial of GA(x,y)GB conforms to the relation [22,23]  Since GA and GB are assumed to be NSSD, . Thus, GA(x,y)GB is nonsingular.
The subgraph GA(x,y)GBx is singular because one of its components is GAx, which is singular since GA is assumed to be NSSD.By the same reasoning, GA(x,y)GBy is also singular.
Let z be any vertex of GA(x,y)GB, different from x and y.Without loss of generality, assume that z is a vertex of GA.

DISCUSSION AND CONCLUDING REMARKS
Our findings indicate that non-bipartite NSSD graphs and molecular graphs are relatively rare objects.Yet, by means of Transformations 1-4, infinitely many molecular graphs of this kind can be constructed.It is worth noting that all our construction methods produce graphs having an even number of vertices.

Figure 1 .
Figure 1.Molecular graphs of some typical alternant hydrocarbons that have no non-bonding molecular orbitals.These all are NSSD graphs.

Figure 2 .
Figure 2.Among 1006674967 non-bipartite graphs with 11 vertices, only 15 are NSSD.None of these are molecular graphs.

Figure 2 .
Figure 2.Among 1006674967 non-bipartite graphs with 11 vertices, only 15 are NSSD.None of these are molecular graphs.

Figure 3 . 1 G
Figure 3.The non-bipartite molecular NSSD graphs with 6 and 8 vertices.Note that 2 G and 3 G are constructed from 1 G by means of Transformation 2 and 4, respectively.

Figure 4 .
Figure 4. Some non-bipartite molecular NSSD graphs with 10 vertices.The species constructed by Transformation 1 (namely,[5]radialene) and those constructed by making use of Transformation 2 are not depicted (but see Figure6).The graph G6 is constructed by Transformation 3 (cf.Figure7).The vertex degrees of the NSSD graphs 1516 17  , , , G G G

Figure 7
Figure 4. Some non-bipartite molecular NSSD graphs with 10 vertices.The species constructed by Transformation 1 (namely,[5]radialene) and those constructed by making use of Transformation 2 are not depicted (but see Figure6).The graph G6 is constructed by Transformation 3 (cf.Figure7).The vertex degrees of the NSSD graphs 1516 17  , , , G G G 18 19 20 Croat.Chem.Acta 2016, 89(4) DOI: 10.5562/cca2997 CONSTRUCTING NON-BIPARTITE NSSD GRAPHS In this section, we describe a few general methods for constructing non-bipartite NSSD graphs.Transformation 1.Let G be any graph with n vertices.Let G* be a graph with 2n vertices, obtained by attaching a new pendent vertex to each vertex of G. Then G*is an NSSD graph.

Figure 7 .
Figure 7. Applying Transformation 3 to construct new nonbipartite NSSD molecular graphs from those obtained by means of Transformation 2.

Figure 8 .
Figure 8. Applying Transformation 4 to construct new non-bipartite NSSD molecular graphs from those obtained by means of Transformation 1.

Figure 9 .
Figure 9. NSSD molecular graphs without pendent vertices, constructed by means of Transformations 2 and 3 from the graphs 4 G and 5 G from Figure 3.

Table 1 .
Number of connected n-vertex graphs