Iskanje po repozitoriju

Opis: The connected sum Z of two root graphs of order n is obtained by gluing them together along a common subgraph G of order n − 1. The two vertices of Z not in G are called terminal vertices. The edged connected sum Z + e is obtained from Z by adding the edge joining the terminal vertices. We consider the case when the root graphs have the same μ–eigenspace of the 0–1 adjacency matrix of dimension one. We show that the μ–eigenspace imposes structural constraints on Z and Z + e, depending on the type of the two vertices. For μ = 0, we investigate the electrical behaviour of a molecular electronic device with structure Z or Z + e, connected at the terminal vertices in a circuit across a small bias voltage. It transpires that the device will be a conductor or insulator depending on the type of the terminal vertices in the 0–eigenspace. We show that conductivity or its barring distinguishes between Z and Z + e.
Ključne besede: molecular electronic device (MED), common eigenspace, μ–core vertices, connected–sum
Objavljeno v RUP: 22.12.2025; Ogledov: 136; Prenosov: 0
Celotno besedilo (398,14 KB)

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: 142; Prenosov: 1
Celotno besedilo (843,13 KB)

Opis: Let A pabe a nonempty subset of positive integers. In this paper we study the set of numerical semigroups that fulfill: if {x,y} ⊆ ℕ\S and x > y > min(S\{0}), then x-y ∉ A.
Ključne besede: Frobenius pseudo-varieties, genus number, numerical semigroups, PD(A)-semigroup and tree (associated to a PD(A)-semigroup)
Objavljeno v RUP: 21.12.2025; Ogledov: 175; Prenosov: 1
Celotno besedilo (356,42 KB)

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: 180; Prenosov: 1
Celotno besedilo (1,11 MB)

Opis: We consider the following question by Balister, Győri and Schelp: given 2^{n-1} nonzero vectors in F_2^n with zero sum, is it always possible to partition the elements of F_2^n into pairs such that the difference between the two elements of the i-th pair is equal to the i-th given vector for every i? An analogous question in F_p, which is a case of the so-called "seating couples" problem, has been resolved by Preissmann and Mischler in 2009. In this paper, we prove the conjecture in F_2^n in the case when the number of distinct values among the given difference vectors is at most n-2log(n)-1, and also in the case when at least a fraction 1/2+ε of the given vectors are equal (for all ε>0 and n sufficiently large based on ε).
Ključne besede: binary vector spaces, seating couples, prescribed differences, perfect matching, functional batch code, graph colourings
Objavljeno v RUP: 21.12.2025; Ogledov: 162; Prenosov: 0
Celotno besedilo (467,06 KB)