1. An infinite family of simple graphs underlying chiral, orientable reflexible and non-orientable rotary mapsIsabel Hubard, Primož Potočnik, Primož Šparl, 2026, original scientific article Abstract: In this paper, we provide the first known infinite family of simple graphs, each of which is the skeleton of a chiral map, a skeleton of a reflexible map on an orientable surface, as well as a skeleton of a reflexible map on a non-orientable surface. This family consists of all lexicographic products $C_n[mK_1]$, where $m \ge 3, n=sm$, with $s$ an integer not divisible by $4$. This answers a question posed by Wilson in 2002.
Keywords: rotary maps, chiral maps, orientable reflexible maps, non-orientable rotary maps
Published in DiRROS: 17.10.2025; Views: 238; Downloads: 121
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3. Edge-transitive cubic graphs: analysis, cataloguing and enumerationMarston D. E. Conder, Primož Potočnik, 2026, original scientific article Abstract: This paper deals with finite cubic (3-regular) graphs whose automorphism group acts transitively on the edges of the graph. Such graphs split into two broad classes, namely arc-transitive and semisymmetric cubic graphs, and then these divide respectively into 7 types (according to a classification by Djoković and Miller (1980) [17]) and 15 types (according to a classification by Goldschmidt (1980) [23]), in terms of certain group amalgams. Such graphs of small order were previously known up to orders 2048 and 768, respectively, and we have extended each of the two lists of all such graphs up to order 10000. Before describing how we did that, we carry out an analysis of the 22 amalgams, to show which of the finitely-presented groups associated with the 15 Goldschmidt amalgams can be faithfully embedded in one or more of the other 21 (as subgroups of finite index), complementing what is already known about such embeddings of the 7 Djoković-Miller groups in each other. We also give an example of a graph of each of the 22 types, and in most cases, describe the smallest such graph, and we then use regular coverings to prove that there are infinitely many examples of each type. Finally, we discuss the asymptotic enumeration of the graph orders, proving that if $f_{\mathcal C}(n)$ is the number of cubic edge-transitive graphs of type ${\mathcal C}$ on at most $n$ vertices, then there exist positive real constants $a$ and $b$ and a positive integer $n_0$ such that $n^{a \log(n)} \le f_{\mathcal C}(n) \le n^{b \log(n)}$ for all $n\ge 0$.
Keywords: groups, graphs, symmetry, amalgams, cover
Published in DiRROS: 21.08.2025; Views: 331; Downloads: 184
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4. Normal covers of 2-arc-transitive graphs of prime-power orderMarston D. E. Conder, Primož Potočnik, 2025, original scientific article Abstract: In a paper by Cai Heng Li in Bull. London Math. Soc. 33 (2001), it was suggested that ‘non-basic’ $2$-arc-transitive graphs of prime-power order that occur as normal covers of smaller $2$-arc-transitive graphs might be rare and difficult to construct. This note describes some of the background to Li’s suggestion, and gives some examples of small valency, and then goes on to show that in fact there are infinitely many examples of valency $d$, for every integer $d \ge 2$. It is also noted that the hypercubes $(Q_n)$ for $n \ge 4$ even, together with a $2$-arc-transitive group $G$ of index $2$ in ${\rm Aut}(Q_n)$, show that the claims of Corollary 1.2 in the above paper by Li are not quite correct.
Keywords: graphs, s-arc-transitive, normal cover
Published in DiRROS: 07.07.2025; Views: 382; Downloads: 238
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5. Vertex-transitive graphs with small motion and transitive permutation groups with small minimal degreeAntonio Montero, Primož Potočnik, 2025, original scientific article Abstract: The motion of a graph is the minimum number of vertices that are moved by a non-trivial automorphism. Equivalently, it can be defined as the minimal degree of its automorphism group (as a permutation group on the vertices). In this paper, we develop some results on permutation groups (primitive and imprimitive) with small minimal degree. As a consequence of such results, we classify vertex-transitive graphs whose motion is 4 or a prime number.
Keywords: fixity, motion, minimal degree, graphs, permutation groups
Published in DiRROS: 26.05.2025; Views: 576; Downloads: 287
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6. Vertex-primitive digraphs with large fixityMarco Barbieri, Primož Potočnik, 2024, original scientific article Abstract: The relative fixity of a digraph $\Gamma$ is defined as the ratio between the largest number of vertices fixed by a nontrivial automorphism of $\Gamma$ and the number of vertices of $\Gamma$ .We characterize the vertex-primitive digraphs whose relative fixity is at least $1 \over 3$, and we show that there are only finitely many vertex-primitive digraphs of bounded out-valency and relative fixity exceeding a positive constant.
Keywords: vertex-primitive digraphs, fixity, product action, graphs
Published in DiRROS: 03.10.2024; Views: 786; Downloads: 511
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7. Cubic vertex-transitive graphs admitting automorphisms of large orderPrimož Potočnik, Micael Toledo, 2023, original scientific article Abstract: A connected graph of order $n$ admitting a semiregular automorphism of order $n/k$ is called a $k$-multicirculant. Highly symmetric multicirculants of small valency have been extensively studied, and several classification results exist for cubic vertex- and arc-transitive multicirculants. In this paper, we study the broader class of cubic vertex-transitive graphs of order $n$ admitting an automorphism of order $n/3$ or larger that may not be semiregular. In particular, we show that any such graph is either a $k$-multicirculant for some $k \le 3$, or it belongs to an infinite family of graphs of girth $6$.
Keywords: cubic vertex-transitive graphs, multicirculants, automorphisms of large order
Published in DiRROS: 18.03.2024; Views: 1045; Downloads: 625
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8. On orders of automorphisms of vertex-transitive graphsPrimož Potočnik, Micael Toledo, Gabriel Verret, 2024, original scientific article Abstract: In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $\Gamma$, $\Gamma \not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\langle g \rangle$ of length equal to the order of $g$. Moreover, we prove that in this case either $\Gamma$ belongs to a well understood family of exceptional graphs or at least $5/12$ of the vertices of $\Gamma$ belong to a regular orbit of $g$. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms $C$ of a connected $3$-valent vertex-transitive graph $\Gamma$ in terms of the number of vertices of $\Gamma$ and the length of a longest orbit of $C$.
Keywords: graphs, automorphism groups, vertex-transitive, regular orbit, cubic, tetravalent
Published in DiRROS: 19.02.2024; Views: 1241; Downloads: 665
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9. A comparison of models for forecasting the residential natural gas demand of an urban areaRok Hribar, Primož Potočnik, Jurij Šilc, Gregor Papa, 2019, original scientific article Abstract: Forecasting the residential natural gas demand for large groups of buildings is extremely important for efficient logistics in the energy sector. In this paper different forecast models for residential natural gas demand of an urban area were implemented and compared. The models forecast gas demand with hourly resolution up to 60 h into the future. The model forecasts are based on past temperatures, forecasted temperatures and time variables, which include markers for holidays and other occasional events. The models were trained and tested on gas-consumption data gathered in the city of Ljubljana, Slovenia. Machine-learning models were considered, such as linear regression, kernel machine and artificial neural network. Additionally, empirical models were developed based on data analysis. Two most accurate models were found to be recurrent neural network and linear regression model. In realistic setting such trained models can be used in conjunction with a weather-forecasting service to generate forecasts for future gas demand.
Keywords: demand forecasting, buildings, energy modeling, forecast accuracy, machine learning
Published in DiRROS: 15.03.2019; Views: 3371; Downloads: 1557
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