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Abstract: We propose a formal model of a knowledge graph (abbr. KG) that classifies the ground triples into sets that correspond to the triple types. The triple types are partially ordered by the sub-type relation. Consequently, the sets of ground triples that are the interpretations of triple types are partially ordered by the subsumption relation. The types of triple patterns restrict the sets of ground triples, which need to be addressed in the evaluation of triple patterns, to the interpretation of the types of triple patterns. Therefore, a schema graph of a KG should include all triple types that are likely to be determined as the types of triple patterns. The stored schema graph consists of the selected triple types that are stored in a KG and the complete schema graph includes all valid triple types of KG. We propose choosing the schema graph, which consists of the triple types from a strip around the stored schema graph, i.e., the triple types from the stored schema graph and some adjacent levels of triple types with respect to the sub-type relation. Given a selected schema graph, the statistics are updated for each ground triple t from a KG. First, we determine the set of triple types stt from the schema graph that are affected by adding a triple t to an RDF store. Finally, the statistics of triple types from the set stt are updated.
Keywords: knowledge graphs, RDF stores, graph database systems
Published in RUP: 16.01.2026; Views: 139; Downloads: 2
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Abstract: A nut graph is a nontrivial graph whose adjacency matrix has a one-dimensional null space spanned by a vector without zero entries. Recently, it was shown that a nut graph has more edge orbits than vertex orbits. It was also shown that for any even $r \geq 2$ and any $k \geq r + 1$, there exist infinitely many nut graphs with r vertex orbits and k edge orbits. Here, we extend this result by finding all the pairs $(r, k)$ for which there exists a nut graph with $r$ vertex orbits and $k$ edge orbits. In particular, we show that for any $k \geq 2$, there are infinitely many Cayley nut graphs with $k$ edge orbits and $k$ arc orbits.
Keywords: nut graph, vertex orbit, edge orbit, arc orbit, Cayley graph, automorphism
Published in RUP: 09.01.2026; Views: 188; Downloads: 5
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Abstract: 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.
Keywords: chemical graph theory, fullerenes, benzenoids, Clar, Fries, Kekule, perfect matching
Published in RUP: 22.12.2025; Views: 154; Downloads: 1
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Abstract: 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.
Keywords: chemical graph theory, anionic Clar number, fullerenes
Published in RUP: 21.12.2025; Views: 200; Downloads: 1
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Abstract: The toughness t(G) of graph G is formalized as the minimum ratio of |S| and ω(G − S) over all vertex subsets S subject to ω(G − S) > 1. As the unique variant parameter of toughness, τ(G) is formulated as the minimum ratio of |S| and ω(G − S) − 1 traversing all the vertex subset S restricted to ω(G − S) ≥ 2. The extant contributions reveal that there is a substantial correlation between toughness and fractional factors. However, there is still a paucity of solid studies on toughness variants τ(G). This work provides several theoretical underpinnings for the tight toughness variant bound for a graph G which admits a fractional k-factor. To be specific, a graph G has a fractional k-factor if τ(G) > k for k ≥ 3 and if τ(G)>3/2 for k = 2. The sharpness of the given bounds is explained by counterexamples.
Keywords: graph, toughness, toughness variant, fractional k-factor
Published in RUP: 21.12.2025; Views: 197; Downloads: 0
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