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11.
Divergence zero quaternionic vector fields and Hamming graphs
Jasna Prezelj, Fabio Vlacci, 2020, izvirni znanstveni članek

Opis: We give a possible extension of the definition of quaternionic power series, partial derivatives and vector fields in the case of two (and then several) non commutative (quaternionic) variables. In this setting we also investigate the problem of describing zero functions which are not null functions in the formal sense. A connection between an analytic condition and a graph theoretic property of a subgraph of a Hamming graph is shown, namely the condition that polynomial vector field has formal divergence zero is equivalent to connectedness of subgraphs of Hamming graphs ▫$H(d, 2)$▫. We prove that monomials in variables ▫$z$▫ and ▫$w$▫ are always linearly independent as functions only in bidegrees ▫$(p, 0)$▫, ▫$(p, 1)$▫, ▫$(0, q)$▫, ▫$(1, q)$▫ and ▫$(2, 2)$▫.
Ključne besede: quaternionic power series, bidegree full functions, Hamming graph, linearly independent quaternionic monomials
Objavljeno v RUP: 03.01.2022; Ogledov: 838; Prenosov: 16
.pdf Celotno besedilo (354,73 KB)

12.
Classification of cubic vertex-transitive tricirculants
Primož Potočnik, Micael Toledo, 2020, izvirni znanstveni članek

Ključne besede: graph, cubic, semiregular automorphism, tricirculant, vertex-transitive
Objavljeno v RUP: 03.01.2022; Ogledov: 802; Prenosov: 40
.pdf Celotno besedilo (1,18 MB)

13.
Splittable and unsplittable graphs and configurations
Nino Bašić, Jan Grošelj, Branko Grünbaum, Tomaž Pisanski, 2019, izvirni znanstveni članek

Opis: We prove that there exist infinitely many splittable and also infinitely many unsplittable cyclic ▫$(n_3)$▫ configurations. We also present a complete study of trivalent cyclic Haar graphs on at most 60 vertices with respect to splittability. Finally, we show that all cyclic flag-transitive configurations with the exception of the Fano plane and the Möbius-Kantor configuration are splittable.
Ključne besede: configuration of points and lines, unsplittable configuration, unsplittable graph, independent set, Levi graph, Grünbaum graph, splitting type, cyclic Haar graph
Objavljeno v RUP: 03.01.2022; Ogledov: 845; Prenosov: 19
.pdf Celotno besedilo (355,79 KB)

14.
A note on acyclic number of planar graphs
Mirko Petruševski, Riste Škrekovski, 2017, izvirni znanstveni članek

Opis: The acyclic number ▫$a(G)$▫ of a graph ▫$G$▫ is the maximum order of an induced forest in ▫$G$▫. The purpose of this short paper is to propose a conjecture that ▫$a(G)\geq \left( 1-\frac{3}{2g}\right)n$▫ holds for every planar graph ▫$G$▫ of girth ▫$g$▫ and order ▫$n$▫, which captures three known conjectures on the topic. In support of this conjecture, we prove a weaker result that ▫$a(G)\geq \left( 1-\frac{3}{g} \right)n$▫ holds. In addition, we give a construction showing that the constant ▫$\frac{3}{2}$▫ from the conjecture cannot be decreased.
Ključne besede: induced forest, acyclic number, planar graph, girth
Objavljeno v RUP: 03.01.2022; Ogledov: 895; Prenosov: 16
.pdf Celotno besedilo (227,50 KB)

15.
Vertex-transitive graphs and their arc-types
Marston D. E. Conder, Tomaž Pisanski, Arjana Žitnik, 2017, izvirni znanstveni članek

Opis: Let ▫$X$▫ be a finite vertex-transitive graph of valency ▫$d$▫, and let ▫$A$▫ be the full automorphism group of ▫$X$▫. Then the arc-type of ▫$X$▫ is defined in terms of the sizes of the orbits of the stabiliser ▫$A_v$▫ of a given vertex ▫$v$▫ on the set of arcs incident with ▫$v$▫. Such an orbit is said to be self-paired if it is contained in an orbit ▫$\Delta$▫ of ▫$A$▫ on the set of all arcs of v$X$▫ such that v$\Delta$▫ is closed under arc-reversal. The arc-type of ▫$X$▫ is then the partition of ▫$d$▫ as the sum ▫$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s)$▫, where ▫$n_1, n_2, \dots, n_t$▫ are the sizes of the self-paired orbits, and ▫$m_1,m_1, m_2,m_2, \dots, m_s,m_s$▫ are the sizes of the non-self-paired orbits, in descending order. In this paper, we find the arc-types of several families of graphs. Also we show that the arc-type of a Cartesian product of two "relatively prime" graphs is the natural sum of their arc-types. Then using these observations, we show that with the exception of ▫$1+1$▫ and ▫$(1+1)$▫, every partition as defined above is \emph{realisable}, in the sense that there exists at least one vertex-transitive graph with the given partition as its arc-type.
Ključne besede: symmetry type, vertex-transitive graph, arc-transitive graph, Cayley graph, cartesian product, covering graph
Objavljeno v RUP: 03.01.2022; Ogledov: 768; Prenosov: 18
.pdf Celotno besedilo (475,17 KB)

16.
Mathematical aspects of fullerenes
Vesna Andova, František Kardoš, Riste Škrekovski, 2016, izvirni znanstveni članek

Opis: Fullerene graphs are cubic, 3-connected, planar graphs with exactly 12 pentagonal faces, while all other faces are hexagons. Fullerene graphs are mathematical models of fullerene molecules, i.e., molecules comprised only by carbon atoms different than graphites and diamonds. We give a survey on fullerene graphs from our perspective, which could be also considered as an introduction to this topic. Different types of fullerene graphs are considered, their symmetries, and construction methods. We give an overview of some graph invariants that can possibly correlate with the fullerene molecule stability, such as: the bipartite edge frustration, the independence number, the saturation number, the number of perfect matchings, etc.
Ključne besede: fullerene, cubic graph, planar graph, topological indices
Objavljeno v RUP: 03.01.2022; Ogledov: 719; Prenosov: 17
.pdf Celotno besedilo (626,25 KB)

17.
Mathematical aspects of Wiener index
Martin Knor, Riste Škrekovski, Aleksandra Tepeh, 2016, izvirni znanstveni članek

Opis: The Wiener index (i.e., the total distance or the transmission number), defined as the sum of distances between all unordered pairs of vertices in a graph, is one of the most popular molecular descriptors. In this article we summarize some results, conjectures and problems on this molecular descriptor, with emphasis on works we were involved in.
Ključne besede: Wiener index, total distance, topological index, molecular descriptor, chemical graph theory
Objavljeno v RUP: 03.01.2022; Ogledov: 978; Prenosov: 27
.pdf Celotno besedilo (434,58 KB)

18.
On colour-preserving automorphisms of Cayley graphs
Ademir Hujdurović, Klavdija Kutnar, Dave Witte Morris, Joy Morris, 2016, izvirni znanstveni članek

Opis: We study the automorphisms of a Cayley graph that preserve its natural edge-colouring. More precisely, we are interested in groups ▫$G$▫, such that every such automorphism of every connected Cayley graph on ▫$G$▫ has a very simple form: the composition of a left-translation and a group automorphism. We find classes of groups that have the property, and we determine the orders of all groups that do not have the property. We also have analogous results for automorphisms that permute the colours, rather than preserving them.
Ključne besede: Cayley graph, automorphism, colour-preserving, colour-permuting
Objavljeno v RUP: 03.01.2022; Ogledov: 682; Prenosov: 16
.pdf Celotno besedilo (412,93 KB)

19.
Testing whether the lifted group splits
Rok Požar, 2016, izvirni znanstveni članek

Opis: Let a group of automorphisms lift along a regular covering projection of connected graphs given combinatorially by means of voltages. The data that determine the lifted group and its action are then conveniently encoded in terms of voltages as well. Along these lines, an algorithm for testing whether the lifted group is a split extension of the group of covering transformations has recently been proposed in the case when the group of covering transformations is solvable. It consists of decomposing the covering into a series of coverings with elementary abelian groups of covering transformations, and inductively solving the problem at every elementary abelian step. Although the explicit construction of the lifted group is not needed, it still involves time and space consuming constructions of certain subgroups in the lifted group at every step except at the final one. In this paper, an improved version that completely avoids such constructions is presented. From voltage distribution we first compute the weak action and the factor set that determine the lifted group, and we then carry out the test by extracting the necessary information only from the corresponding weak actions and factor sets at every step. An experimental comparison is made against the previous version.
Ključne besede: algorithm, graph, group extension, lifting automorphisms, regular covering projection, voltages
Objavljeno v RUP: 03.01.2022; Ogledov: 707; Prenosov: 17
.pdf Celotno besedilo (317,95 KB)

20.
On minimal forbidden subgraphs for the class of EDM-graphs
Gašper Jaklič, Jolanda Modic, 2015, izvirni znanstveni članek

Opis: In this paper, a relation between graph distance matrices and Euclidean distance matrices (EDM) is considered. Graphs, for which the distance matrix is not an EDM (NEDM-graphs), are studied. All simple connected non-isomorphic graphs on ▫$n \le 8$▫ nodes are analysed and a characterization of the smallest NEDM-graphs, i.e., the minimal forbidden subgraphs, is given. It is proven that bipartite graphs and some subdivisions of the smallest NEDM-graphs are NEDM-graphs, too.
Ključne besede: graph theory, graph, Euclidean distance matrix, distance, eigenvalue
Objavljeno v RUP: 31.12.2021; Ogledov: 860; Prenosov: 18
.pdf Celotno besedilo (711,65 KB)

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