1. Wandering domains arising from Lavaurs maps with Siegel disksMatthieu Astorg, Luka Boc Thaler, Han Peters, 2023, original scientific article Abstract: The first example of polynomial maps with wandering domains was constructed in 2016 by the first and last authors, together with Buff, Dujardin and Raissy. In this paper, we construct a second example with different dynamics, using a Lavaurs map with a Siegel disk instead of an attracting fixed point. We prove a general necessary and sufficient condition for the existence of a trapping domain for nonautonomous compositions of maps converging parabolically towards a Siegel-type limit map. Constructing a skew-product satisfying this condition requires precise estimates on the convergence to the Lavaurs map, which we obtain by a new approach. We also give a self-contained construction of parabolic curves, which are integral to this new method.
Keywords: Fatou sets, holomorphic dynamics, parabolic implosion, polynomial mappings, skew-products, wandering Fatou components, parabolic curves, nonautonomous dynamics
Published in DiRROS: 09.04.2024; Views: 135; Downloads: 48
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2. A method for computing the edge-Hosoya polynomial with application to phenylenesMartin Knor, Niko Tratnik, 2023, original scientific article Abstract: The edge-Hosoya polynomial of a graph is the edge version of the famous Hosoya polynomial. Therefore, the edge-Hosoya polynomial counts the number of (unordered) pairs of edges at distance $k \ge 0$ in a given graph. It is well known that this polynomial is closely related to the edge-Wiener index and the edge-hyper-Wiener index. As the main result of this paper, we greatly generalize an earlier result by providing a method for calculating the edge-Hosoya polynomial of a graph $G$ which is obtained by identifying two edges of connected bipartite graphs $G_1$ and $G_2$. To show how the main theorem can be used, we apply it to phenylene chains. In particular, we present the recurrence relations and a linear time algorithm for calculating the edge-Hosoya polynomial of any phenylene chain. As a consequence, closed formula for the edge-Hosoya polynomial of linear phenylene chains is derived.
Keywords: edge-Hosoya polynomial, graphs, phenylenes
Published in DiRROS: 18.03.2024; Views: 271; Downloads: 325
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3. How to compute the M-polynomial of (chemical) graphsEmeric Deutsch, Sandi Klavžar, Gašper Domen Romih, 2023, original scientific article Abstract: Let $G$ be a graph and let $m_{i,j}(G)$, $i,j\ge 1$, be the number of edges $uv$ of ▫$G$▫ such that $\{d_v(G), d_u(G)\} = \{i,j\}$. The M-polynomial of $G$ is $M(G;x,y) = \sum_{i\le j} m_{i,j}(G)x^iy^j$. A general method for calculating the M-polynomials for arbitrary graph families is presented. The method is further developed for the case where the vertices of a graph have degrees 2 and $p$, where $p\ge 3$, and further for such planar graphs. The method is illustrated on families of chemical graphs.
Keywords: M-polynomial, chemical graph, planar graph
Published in DiRROS: 18.03.2024; Views: 106; Downloads: 38
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4. Invariants of multi-linkoidsBoštjan Gabrovšek, Neslihan Gügümcü, 2023, original scientific article Abstract: In this paper, we extend the definition of a knotoid to multilinkoids that consist of a finite number of knot and knotoid components. We study invariants of multi-linkoids, such as the Kauffman bracket polynomial, ordered bracket polynomial, the Kauffman skein module, and the $T$-invariant in relation with generalized $\Theta$-graphs.
Keywords: knotoid, multi-linkoid, spatial graph, Kauffman bracket polynomial, Kauffman bracket skein module, theta-curve, theta-graph
Published in DiRROS: 15.03.2024; Views: 114; Downloads: 66
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5. General position polynomialsVesna Iršič, Sandi Klavžar, Gregor Rus, James Tuite, 2024, original scientific article Abstract: A subset of vertices of a graph $G$ is a general position set if no triple of vertices from the set lie on a common shortest path in $G$. In this paper we introduce the general position polynomial as $\sum_{i \geq 0} a_i x^i$, where $a_i$ is the number of distinct general position sets of $G$ with cardinality $i$. The polynomial is considered for several well-known classes of graphs and graph operations. It is shown that the polynomial is not unimodal in general, not even on trees. On the other hand, several classes of graphs, including Kneser graphs $K(n,2)$, with unimodal general position polynomials are presented.
Keywords: general position set, general position number, general position polynomial, unimodality, trees, Cartesian product of graphs, Kneser graphs
Published in DiRROS: 28.02.2024; Views: 140; Downloads: 95
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