Symmetry, Graph Reconstruction and Molecular Conduction

Dear Colleagues,

Algebraic combinatorics and spectral graph theory present the ideal interface between linear algebra and quantum molecular chemistry [10]. Research in basic mathematical theory and its applications offers amazing mutual benefits to both areas. Ulam’s reconstruction conjecture (RC), posed in 1942, is still open. O’Neill describes it as a mathematical rarity, as the problem is easily stated [1,2]. Anyone can take a crack at it, but it is certainly not trivial. A graph G (sometimes called a network) is a finite set V of n vertices with a set of edges connecting selected pairs of vertices. A deck D of cards for G is a multi-set of n unlabeled subgraphs G - v on n - 1 vertices, obtained from G by deleting a vertex v, at a time, with its incident edges. Ulam claimed that the parent graph G can be reconstructed from D. Symmetry in a graph plays an important role in its reconstruction, as repeated cards tend to show up in D for graphs with non-trivial automorphism groups. Other important variants of the RC have been proposed, most of which are still open problems [7]. One is polynomial reconstruction (PR). Simply stated, the PR problem attempts to retrieve the eigenvalues of the adjacency matrix of G from the polynomial deck (PD) of n cards, each containing eigenvalues of G - v, as v runs through the vertices of V.

The PR problem proves to be addictive to many graph theorists working in the field of spectral graph theory [3,4,5,6,8,9]. The RC and its variants may be considered to fall under the category of the renowned isomorphism problem that attempts to establish a combination of combinatorial and algebraic parameters that determine a graph uniquely. The techniques used are varied and sought after in the recovery of missing data in various scientific areas. Important theorems that are developed in the processes leading to the reconstruction of a parent graph or of its parameters have shed light on the physical and chemical behavior of Pi molecular systems, such as fullerenes and graphene-like sheets. Spectral graph theoretical results, both well known and newly developed, have proved to fit like a glove to clarify poorly understood quantum mechanical processes, particularly in relation to ballistic electron conduction in molecules [14,18].

The mechanical, chemical, and electronic properties of materials change as their size approaches the nanoscale, where the percentage of atoms at the surface of the material becomes significant. Mesoscopic systems, which are between the atomic and macroscopic scales, exhibit surprising novel phenomena [20]. There has been great interest in molecular systems recently, as they find numerous applications in the creation of electronic devices. They are also proving to be useful in the development of spin electronics in solid-state devices, i.e., the study of the intrinsic spin of the electron and its associated magnetic moment, in addition to its fundamental electronic charge. The transition from large systems to smaller ones can be influenced by mesoscopic effects, which provides another aspect within the area of mesoscopic physics.

Within the source-sink potential (SSP) model, the tight binding equations for ballistic conduction through conjugated molecular structures provide varying notions to explain current flow through a molecular device with two atom terminals. This led to the discovery of omni-conducting and omni-insulating devices, where the electron flow or lack of it is independent of the atom-connecting terminals [13,19].

The recent increase of interest in the photo-magnetization of Möbius aromatic molecules stems from their particular structure and from claims that they could have innovative applications [21,22,23]. In the laboratory, they are being studied for their potential to be applied in organic solar batteries, lights, and conductive materials. The carbon framework of a Möbius network is embedded on a surface with a half twist. It can be represented by a signed graph, where negatively weighted edges represent a twist. Such molecules have been synthesized, and they obey different electron counting rules from those of the standard unweighted Hückel molecular graphs [19]. The lack of understanding of experimental observations when such a molecule is energized is calling for the creation of new mathematical theory in the field of signed adjacency matrices of the graphs representing the molecular skeleton.

The aim of the present Special Issue is to highlight the wealth of graph spectra as a basic mathematical theory [11,12,15,16,17,24,25,26,27]. The need to back physical, chemical, biological, AI and statistical observations with solid mathematical reasons tends to promote the development of novel mathematical theory. Thus, all fields stand to gain from this interplay with the most creative artistic science, mathematics. We are soliciting contributions (research and review articles) covering a broad range of topics on graph theory, symmetry and molecular conductivity, including (though not limited to) the following.

References

[1] F. Harary. A survey of the reconstruction conjecture. In: R.A. Bari, F. Harary F. (eds) Graphs and Combinatorics. Lecture Notes in Mathematics, vol 406. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0066431 Graphs and Combinatorics pp 18-28, 1974.

[2] P.V. O’Neil. Ulam’s Conjecture and Graph Reconstructions. The American Mathematical Monthly, 77:1, 35-43, 1970. DOI: 10.1080/00029890.1970.11992413

[3] I. Sciriha and M. J. Formosa. On polynomial reconstruction of disconnected graphs. Util. Math., 64:33–44, 2003.

[4] I. Sciriha. Polynomial reconstruction and graphs with a singular deck. The Cyprus Journal of Sciences, 1, 2002.

[5] I. Sciriha. Polynomial reconstruction and terminal vertices. Linear Algebra Appl., 356:145–156, 2002. Special issue on algebraic graph theory (Edinburgh, 2001).

[6] I. Sciriha. Polynomial reconstruction: old and new techniques. Rend. Sem. Mat. Messina Ser. II, 8(24)(suppl.):163–179, 2002. 6thWorkshop on Combinatorics (Messina, 2002).

[7] K.J. Asciak, M.A. Francalanza, J. Lauri, W. Myrvold. A survey of some open questions in reconstruction numbers. Article in Ars Combinatoria -Waterloo then Winnipeg - October 2010 Cambridge University Press, 2016. 194, 2016.

[8] J. Coates, J. Lauri, and I. Sciriha. Polynomial reconstruction for certain subclasses of disconnected graphs. Graph Theory Notes of New York, LXIII(5):41–48, 2013.

[9] A. Farrugia and I. Sciriha. Non-singular graphs with a singular deck. Discrete Applied Mathematics, 202:50–57, 2016.

[10] I. Sciriha and A. Farrugia. From nutgraphs to molecular structure and conductivity. Mathematical Chemistry Monographs, University of Kragujevac, Series Eds. I. Gutman and B. Furtula, 2021.

[11] I. Sciriha. Coalesced and Embedded Nut Graphs in Singular Graphs. Ars Mathematica Contemporanea, 1:20–31 (http://amc.imfm.si), 2008.

[12] A. J. Schwenk. On the eigenvalues of a graph. In L.W. Beineke and R. J.Wilson, editors, Selected Topics in Graph Theory, chapter 11, pages 307–336. Academic Press, 1978.

[13] M.C. Petty, T. Nagase, H. Suzuki, and H. Naito. Molecular Electronics. In S. Kasap and P. Capper, editors. Springer Handbook of Electronic and Photonic Materials. Springer International Publishing. 2017. https://doi.org/10.1007/978-3-319-48933-951

[14] P. W. Fowler, M. Borg, B. T. Pickup, and I. Sciriha. Molecular graphs and molecular conduction: the d-omni-conductors. Phys. Chem. Chem. Phys., 22(3)::1349–1358, 2019.

[15] I. Sciriha, J. Briffa, and M. Debono. Fast algorithms for indices of nested split graphs approximating real complex networks. Discrete Applied Mathematics, 247:152–164, 2018.

[16] J. Briffa and I. Sciriha. On the displacement of eigenvalues when removing a twin vertex. Discussiones Mathematicae. Graph Theory, 40–2:435–450, 2020.

[17] I Gutman, B. Furtula, A. Farrugia, and I. Sciriha. Constructing nssd molecular graphs. Croat. Chem. Acta, 89(4):449–454, December 2016.

[18] B. T. Pickup, P. W. Fowler, M. Borg, and I. Sciriha. A new approach to the method of source-sink potentials for molecular conduction. The Journal of Chemical Physics, 143(19), 2015.

[19] P. W. Fowler, B. T. Pickup, T. Z. Todorova, M. Borg, and I. Sciriha. Omni-conducting and omni-insulating molecules. The Journal of Chemical Physics, 140(5), 054115, doi:10.1063/1. 4863559. 2014.

[20] I. Sciriha and P. W. Fowler. Nonbonding orbitals in fullerenes: Nuts and

cores in singular polyhedral graphs. J. Chem. Inf. Model. 47 (2007), 1763–1775,

doi:10.1021/ci700097j.

[21] F. Ema, M. Tanabe, S. Saito, T. Yoneda, K. Sugisaki, T. Tachikawa, S. Akimoto, S. Yamauchi, K. Sato, A. Osuka, T. Takui, Y. Kobori. Charge-Transfer Character Drives Möbius Antiaromaticity in the Excited Triplet State of Twisted [28] Hexaphyrin. The Journal of Physical Chemistry Letters 2018(9) 2685–2690, 2018.

[22] R. Herges. Topology in chemistry: Designing Möbius molecules. Chem. Rev. 106 (2006), 4820–4842, doi:10.1021/cr0505425.

[23] H. S. Rzepa. Möbius aromaticity and delocalization. Chem. Rev. 105 3697–3715. 2005. doi:10.1021/cr030092l.

[24] I. Sciriha. Maximal and extremal singular graphs. Sovremennaya Matematika i Ee Prilozheniya-Contemporary Mathematics and its Applications, 71:1–9, 2011. Journal of Mathematical Sciences, 182:2, 2012.

[25] R. Rowlinson. More on graph perturbations. Bull. London Math. Soc., pages 209–216, 1990.

[26] I. M. Gutman D. M. Cvetkovi´c. The algebraic multiplicity of the number zero in the spectrum of a bipartite graph. Matematiˇcki Vesnik, 9(24)(56):141–150, 1972.

[27] Stanley Fiorini, Ivan Gutman, and I. Sciriha. Trees with maximum nullity. Linear, Algebra Appl., 397:245–251, 2005.

Prof. Irene Sciriha
Guest Editor

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• graph reconstruction conjecture and its variants
• the polynomial reconstruction problem
• the recovery of missing data
• spectral properties of graphs
• eigenvector techniques
• the role of symmetry in graphs and their automorphisms
• signed graphs
• efficient algorithms for graphs
• the structure and conductivity of Pi molecular systems
• conductivity in nano and mesoscopic structures
• twisted carbon structures