1. Nut digraphsNino Bašić, Patrick W. Fowler, Maxine M. McCarthy, Primož Potočnik, 2026, izvirni znanstveni članek Opis: A nut graph is a simple graph whose kernel is spanned by a single full vector (i.e., the adjacency matrix has a single zero eigenvalue and all non-zero kernel eigenvectors have no zero entry). We classify generalisations of nut graphs to nut digraphs: a digraph whose kernel (resp. co-kernel) is spanned by a full vector is dextro-nut (resp. laevo-nut); a bi-nut digraph is both laevo- and dextro-nut; an ambi-nut digraph is a bi-nut digraph where kernel and co-kernel are spanned by the same vector; a digraph is inter-nut if the intersection of the kernel and co-kernel is spanned by a full vector. It is known that a nut graph is connected, leafless and non-bipartite. It is shown here that an ambi-nut digraph is strongly connected, non-bipartite (i.e., has a non-bipartite underlying graph) and has minimum in-degree and minimum out-degree of at least 2. Refined notions of core and core-forbidden vertices apply to singular digraphs. Infinite families of nut digraphs and systematic coalescence, crossover and multiplier constructions are introduced. Relevance of nut digraphs to topological physics is discussed.
Ključne besede: nut graph, core graph, nullity, directed graph, nut digraph, dextro-nut, laevo-nut, bi-nut, ambi-nut, inter-nut, dextro-core vertex, laevo-core vertex, graph spectra
Objavljeno v RUP: 09.01.2026; Ogledov: 197; Prenosov: 8
 Celotno besedilo (873,25 KB)
Gradivo ima več datotek! Več... |
2. Clar and Fries structures for fullerenesPatrick W. Fowler, Wendy Myrvold, Rebecca L. Vandenberg, Elizabeth J. Hartung, Jack E. Graver, 2026, izvirni znanstveni članek Opis: Fries and Clar numbers are qualitative indicators of stability in conjugated π systems. For a given Kekulé structure, call any hexagon that contains three double bonds benzenoid. The Fries number is the maximum number of benzenoid hexagons, whereas the Clar number is the maximum number of independent benzenoid hexagons, in each case taken over all Kekulé structures. A Kekulé structure that realises the Fries (Clar) number is a Fries (Clar) structure. For benzenoids, it is not known whether every Fries structure is also a Clar structure. For fullerenes C_n, it is known that some Clar structures in large examples correspond to no Fries structure. We show that Fries structures that are not Clar occur early: examples where some Fries structure is not Clar start at C_34, and examples where no Fries structure is Clar start at C_48. Hence, it is unsafe to use fullerene Fries structures as routes to Clar number. However, Fries structures often describe the neutral fullerene better than a Clar structure, e.g. in rationalising bond lengths in the experimental isomer of C_60. Conversely, an extension of Clar sextet theory suggests the notion of anionic Clar number for fullerene anions, where both pentagons and hexagons may support sextets.
Ključne besede: chemical graph theory, fullerenes, benzenoids, Clar, Fries, Kekule, perfect matching
Objavljeno v RUP: 22.12.2025; Ogledov: 187; Prenosov: 2
Celotno besedilo (843,13 KB) |
3. Tight upper bounds for the p-anionic Clar number of fullerenesAaron Slobodin, Wendy Myrvold, Gary MacGillivray, Patrick W. Fowler, 2026, izvirni znanstveni članek Opis: A fullerene is an all-carbon molecule with a polyhedral structure where each atom is bonded to three other atoms and each face is either a pentagon or a hexagon. Fullerenes correspond to 3-regular planar graphs whose faces have sizes 5 or 6. The p-anionic Clar number C_(p)(G) of a fullerene G is equal to p + h, where h is maximized over all choices of p + h independent faces (exactly p pentagons and h hexagons) the deletion of whose vertices leave a graph with a perfect matching. This definition is motivated by the chemical observation that pentagonal rings can accommodate an extra electron, so that the pentagons of a
fullerene with charge −p, compete with the hexagons to host ‘Clar sextets’ of six electrons, and pentagons will preferentially acquire the p excess electrons of the anion.
Tight upper bounds are established for the p-anionic Clar number of fullerenes for p > 0. The upper bounds are derived via graph theoretic arguments and new results on minimal cyclic-k-edge cutsets in IPR fullerenes (fullerenes that have all pentagons pairwise disjoint). These bounds are shown to be tight by infinite families of fullerenes that achieve them.
Ključne besede: chemical graph theory, anionic Clar number, fullerenes
Objavljeno v RUP: 21.12.2025; Ogledov: 232; Prenosov: 1
Celotno besedilo (1,11 MB) |
4. Nut graphs with a given automorphism groupNino Bašić, Patrick W. Fowler, 2025, izvirni znanstveni članek Opis: A nut graph is a simple graph of order 2 or more for which the adjacency matrix has a single zero eigenvalue such that all nonzero kernel eigenvectors have no zero entry (i.e. are full). It is shown by construction that every finite group can be represented as the group of automorphisms of infinitely many nut graphs. It is further shown that such nut graphs exist even within the class of regular graphs; the cases where the degree is 8, 12, 16, 20 or 24 are realised explicitly.
Ključne besede: nut graph, graph automorphism, automorphism group, nullity, graph spectra, f-universal
Objavljeno v RUP: 25.11.2025; Ogledov: 467; Prenosov: 4
Celotno besedilo (526,71 KB)
Gradivo ima več datotek! Več... |
5. |
6. |
7. Convexity deficit of benzenoidsNino Bašić, Sarah J. Berkemer, Jörg Fallmann, Patrick W. Fowler, Thomas Gatter, Tomaž Pisanski, Nancy Retzlaff, Peter F. Stadler, Sara Sabrina Zemljič, 2019, izvirni znanstveni članek Ključne besede: benzenoid, fusene, convexity deficit, convex benzenoid, quasi-convex benzenoid, pseudo-convex benzenoid
Objavljeno v RUP: 28.05.2020; Ogledov: 3023; Prenosov: 60
 Povezava na celotno besedilo |
