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Abstract: The aim of this paper is to investigate resonance graphs of $2$-connected outerplane bipartite graphs, which include various families of molecular graphs. Firstly, we present an algorithm for a binary coding of perfect matchings of these graphs. Further, $2$-connected outerplane bipartite graphs with isomorphic resonance graphs are considered. In particular, it is shown that if two $2$-connected outerplane bipartite graphs are evenly homeomorphic, then its resonance graphs are isomorphic. Moreover, we prove that for any $2$-connected outerplane bipartite graph $G$ there exists a catacondensed even ring systems $H$ such that the resonance graphs of $G$ and $H$ are isomorphic. We conclude with the characterization of $2$-connected outerplane bipartite graphs whose resonance graphs are daisy cubes.
Keywords: graph theory, resonance graphs, bipartite graphs
Published in DKUM: 10.12.2024; Views: 0; Downloads: 11
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Abstract: Szeged, Padmakar-Ivan (PI), and Mostar indices are some of the most investigated distance-based Szeged-like topological indices. On the other hand, the polynomials related to these topological indices were also introduced, for example the Szeged polynomial, the edge- Szeged polynomial, the PI polynomial, the Mostar polynomial, etc. In this paper, we introduce a concept of the general Szeged-like polynomial for a connected strength-weighted graph. It turns out that this concept includes all the above mentioned polynomials and also infinitely many other graph polynomials. As the main result of the paper, we prove a cut method which enables us to efficiently calculate a Szeged-like polynomial by using the corresponding polynomials of strength-weighted quotient graphs obtained by a partition of the edge set that is coarser than ▫$\Theta^*$▫-partition. To the best of our knowledge, this represents the first implementation of the famous cut method to graph polynomials. Finally, we show how the deduced cut method can be applied to calculate some Szeged-like polynomials and corresponding topological indices of para-polyphenyl chains and carbon nanocones.
Keywords: graph theory, carbon nanocone, topological indices
Published in DKUM: 25.03.2024; Views: 234; Downloads: 5
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Abstract: The booklet contains the abstracts of the talks given at the 30th Ljubljana-Leoben Graph Theory Seminar that was held at the Faculty of Natural Sciences and Mathematics in Maribor between 13-15 September, 2017. The seminar attracted more than 30 participants from eight countries, all of which are researchers in different areas of graph theory. The topics of the talks encompass a wide range of contemporary graph theory research, notably, various types of graph colorings (b-coloring, packing coloring, edge colorings), graph domination (rainbow domination, Grundy domination, graph security), distinguishing problems, algebraic graph theory, graph algorithms, chemical graph theory, coverings, matchings and also some classical extremal problems. Beside the abstracts of the four invited speakers (Csilla Bujtás, Premysl Holub, Jakub Przybyło, Zsolt Tuza), the booklet contains also the abstracts of 18 contributed talks given at the event.
Keywords: mathematics, discrete mathematics, graph theory, Ljubljana-Leoben seminar, Maribor, Slovenia
Published in DKUM: 08.12.2017; Views: 1625; Downloads: 218
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Abstract: Roman domination is a historically inspired variety of general domination such that every vertex is labeled with labels from $\{0,1,2\}$. Roman domination number is the smallest of the sums of labels fulfilling condition that every vertex, labeled 0, has a neighbor, labeled 2. Using algebraic approach we give ▫$O(C)$▫ time algorithm for computing Roman domination number of special classes of polygraphs (rota- and fasciagraphs). By implementing the algorithm we give formulas for Roman domination number of the Cartesian products of paths and cycles ▫$P_n \Box P_k$▫, ▫$P_n \Box C_k$▫ for ▫$k \leq 8$▫ and ▫$n \in {\mathbb N}$▫ and for ▫$C_n \Box P_k$▫ and ▫$C_n \Box C_k$▫ for ▫$k \leq 5$▫, ▫$n \in {\mathbb N}$▫. We also give a list of Roman graphs among investigated families.
Keywords: graph theory, Roman domination number, Cartesian product, polygraphs, path algebra
Published in DKUM: 23.08.2017; Views: 1634; Downloads: 258
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Abstract: In contrast to the widespread interest in the Frame-Stewart conjecture (FSC) about the optimal number of moves in the classical Tower of Hanoi task with more than three pegs, this is the first study of the question of investigating shortest paths in Hanoi graphs ▫$H_p^n$▫ in a more general setting. Here ▫$p$▫ stands for the number of pegs and ▫$n$▫ for the number of discs in the Tower of Hanoi interpretation of these graphs. The analysis depends crucially on the number of largest disc moves (LDMs). The patterns of these LDMs will be coded as binary strings of length ▫$p-1$▫ assigned to each pair of starting and goal states individually. This will be approached both analytically and numerically. The main theoretical achievement is the existence, at least for all ▫$n \geqslant p(p-2)$▫, of optimal paths where ▫$p-1$▫ LDMs are necessary. Numerical results, obtained by an algorithm based on a modified breadth-first search making use of symmetries of the graphs, lead to a couple of conjectures about some cases not covered by our ascertained results. These, in turn, may shed some light on the notoriously open FSC.
Keywords: graph theory, Tower of Hanoi, Hanoi graphs, shortest paths, symmetries, breadth-first search
Published in DKUM: 14.08.2017; Views: 1714; Downloads: 250
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