Raziskovalni matematični seminar - Arhiv

We report on work in progress on the question of finding good paths for bicycles in a traffic network with (possibly coordinated) traffic lights. While the standard dynamic shortest path problem in time-dependent networks with cyclic time windows is known to be easy, we study the related problem of finding routes that keep a small number of stops.  We show that the problem is strongly NP-hard for paths and trails, whereas it is weakly NP-hard for walks. We give a pseudo-polynomial algorithm for this third option. Further, we report about ongoing work on designing algorithms for the case of variable speeds and the employment of appropriate power consumption and recovery models for bicycles for the above formulated problem.

Joint work with Markus Rogge, Robert Scheffler & Martin Strehler.

In this talk I will describe some progress in analytic number theory, from the pioneering work by Dirichlet and Riemann to the Selberg class of L-functions. This talk will be informative and should be accessible to non-specialists.

Let Γ denote a Q-polynomial distance-regular graph with diameter D and valency k ≥ 3. By the result of H. Lewis, the girth of Γ  is at most 6. In this talk, we give a classification of graphs that attain this upper bound. We show that Γ  has girth 6 if and only if it is either isomorphic to the Odd graph on a set of cardinality 2D +1, or to a generalized hexagon of order (1, k -1).

The lecture will present key novelties in the field of Open Science (OS), with a focus on changes at the Slovenian Research and Innovation Agency (ARIS), national and international initiatives, and activities at the University of Primorska (UP). We will review the new ARIS requirements regarding research data management plans, trusted repositories, citizen science, and other topics. Key updates in open access to scientific publications will be presented, including licensing requirements and changes in funding for publication costs. A section will be dedicated to the SPOZNAJ project, which enhances training and support for Open Science in Slovenia, and to the activities of the Slovenian Open Science Community (SSOZ). UP is actively organizing new events and training sessions to promote Open Science, including online lectures, workshops, and professional support for researchers. Finally, we will also introduce the latest Open Science activities within the T4EU university alliance.

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A theorem of Mader states that in every finite (k+1)-edge-connected digraph D, for any s, t ∈ V(D), there exists an st-path P such that D - E(P) remains k-edge-connected. We show that this result does not extend to infinite digraphs and can fail drastically. Specifically, for every k ∈ ℕ, we construct a "fractal-like" infinite k-edge-connected digraph D_k with s, t ∈V(D_k), in which every st-path shares an edge with every ts-path.

A nut graph is a nontrivial simple graph whose adjacency matrix contains a one-dimensional null space spanned by a vector without zero entries. Moreover, an ℓ-circulant graph is a graph that admits a cyclic group of automorphisms having ℓ vertex orbits of equal size. It is not difficult to verify that there is no cubic 1-circulant nut graph or cubic 2-circulant nut graph, while the full classification of cubic 3-circulant nut graphs was recently obtained [Electron. J. Comb. 31(2) (2024), #2.31]. Here, we investigate the existence of cubic ℓ-circulant nut graphs for ℓ ≥ 4 and show that there is no cubic ℓ-circulant nut graph for ℓ ∈ {4, 5}, while there are infinitely many cubic ℓ-circulant nut graphs for each ℓ ∈ {6, 7} or ℓ ≥ 9. We also prove that there are infinitely many d-regular nut graphs for each d ≥ 3.

This is a joint work with Nino Bašić and Patrick W. Fowler.

A nut graph is a simple graph for which the adjacency matrix has a single zero eigenvalue such that all non-zero kernel eigenvectors have no zero entry. Nut graphs have been established in the literature for quite some time; they were introduced in 1998 by Sciriha and Gutman. First, we present some well-known results. Nut graphs have seven or more vertices; they are all connected, non-bipartite, and leafless. Several constructions for nut graphs from smaller starting graphs are known (e.g., the Fowler construction); to these we add multiplier constructions that yield nut graphs from regular graphs (that are not necessarily nut graphs). A class of particular interest is the class of chemical (i.e. subcubic) nut graphs; all $(v_3, v_2)$ pairs, for which a chemical nut graphs with $v_3$ degree-3 and $v_2$ degree-2 vertices exist, were characterised a few years ago.

In this talk, we will focus on certain symmetry properties of nut graphs. First, we show by construction that every finite group can be represented as the group of automorphisms of infinitely many (regular) nut graphs. Next, we show that a nut graph always has at least one more edge orbit than it has vertex orbits. In particular, edge-transitive nut graphs do not exist. We also give infinite families of vertex-transitive nut graphs with two orbits of edges.