1. Resonance graphs of plane bipartite graphs as daisy cubesSimon Brezovnik, Zhongyuan Che, Niko Tratnik, Petra Žigert Pleteršek, 2025, izvirni znanstveni članek Ključne besede: daisy cube, fries number, peripherally 2-colorable, plane (weakly) elementary bipartite graph, resonance graph Objavljeno v DKUM: 31.01.2025; Ogledov: 0; Prenosov: 0
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2. Zagreb root-indices of graphs with chemical applicationsNiko Tratnik, Petra Žigert Pleteršek, 2024, izvirni znanstveni članek Opis: Root-indices of graphs are mathematical tools that help us to understand complex systems, like molecules and networks, by capturing key structural information. In this study, we introduce two new root-indices, the first and the second Zagreb root-index, and we analyze their properties. We apply these indices to chemical structures like benzenoid molecules and octane isomers, showing that they sometimes provide better insights than traditional indices. We also compare the effectiveness of several root-indices with their standard versions, highlighting their ability to distinguish between different graph structures. Ključne besede: first Zagreb index, second Zagreb index, root-indices, octane isomers, discrimination power Objavljeno v DKUM: 19.12.2024; Ogledov: 0; Prenosov: 3
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3. Resonance graphs and a binary coding of perfect matchings of outerplane bipartite graphsSimon Brezovnik, Niko Tratnik, Petra Žigert Pleteršek, 2023, izvirni znanstveni članek Opis: 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. Ključne besede: graph theory, resonance graphs, bipartite graphs Objavljeno v DKUM: 10.12.2024; Ogledov: 0; Prenosov: 6
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4. Weighted wiener indices of molecular graphs with application to alkenes and alkadienesSimon Brezovnik, Niko Tratnik, Petra Žigert Pleteršek, 2021, izvirni znanstveni članek Opis: There exist many topological indices that are calculated on saturated hydrocarbons since they can be easily modelled by simple graphs. On the other hand, it is more challenging to investigate topological indices for hydrocarbons with multiple bonds. The purpose of this paper is to introduce a simple model that gives good results for predicting physico-chemical properties of alkenes and alkadienes. In particular, we are interested in predicting boiling points of these molecules by using the well known Wiener index and its weighted versions. By performing the non-linear regression analysis we predict boiling points of alkenes and alkadienes. Ključne besede: weighted Wiener index, alkenes, alkadienes, boiling point Objavljeno v DKUM: 30.05.2024; Ogledov: 115; Prenosov: 14
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5. Szeged-like entropies of graphsMatthias Dehmer, Frank Emmert-Streib, Niko Tratnik, Petra Žigert Pleteršek, 2022, izvirni znanstveni članek Ključne besede: Szeged entropy, Mostar entropy, PI entropy, cut method, quotient graphs, sensitivity of a topological descriptor Objavljeno v DKUM: 20.05.2024; Ogledov: 164; Prenosov: 14
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7. Outerplane bipartite graphs with isomorphic resonance graphsSimon Brezovnik, Zhongyuan Che, Niko Tratnik, Petra Žigert Pleteršek, 2024, izvirni znanstveni članek Opis: We present novel results related to isomorphic resonance graphs of 2-connected outerplane bipartite graphs. As the main result, we provide a structure characterization for 2-connected outerplane bipartite graphs with isomorphic resonance graphs. Three additional characterizations are expressed in terms of resonance digraphs, via local structures of inner duals, as well as using distributive lattices on the set of order ideals of posets defined on inner faces of 2-connected outerplane bipartite graphs. Ključne besede: distributive lattice, inner dual, isomorphic resonance graphs, order ideal, 2-connected outerplane bipartite graph Objavljeno v DKUM: 29.02.2024; Ogledov: 290; Prenosov: 10
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8. Resonančni grafi nekaterih dvodelnih zunajravninskih grafov in posplošena metoda prerezov : doktorska disertacijaSimon Brezovnik, 2022, doktorska disertacija Opis: V doktorski disertaciji se najprej ukvarjamo z resonančnimi grafi katakondenziranih sodih obročnih sistemov (CERS-ov) in njihovo povezavo z marjetičnimi kockami. V nadaljevanju razvijemo posplošeno metodo prerezov, ki omogoča izračun različnih topoloških indeksov (Wienerjevega indeksa dvojno vozliščno-uteženega grafa, Schultzevega indeksa ter indeksov tipa Szeged). V uvodnem poglavju so predstavljeni nekateri že znani rezultati v povezavi z resonančnimi grafi in posplošeno metodo prerezov. Prav tako v nekaj stavkih napovemo rezultate, ki sledijo v nadaljevanju. V drugem poglavju zapišemo osnovne definicije, ki se dotikajo področja teorije grafov in so potrebne za razumevanje osrednjega dela.
V tretjem poglavju predstavimo vse obravnavane kemijske strukture in grafe, ki modelirajo te strukture. Najprej obravnavamo benzenoidne sisteme, zatem opišemo CERS-e, fenilene in koronoide.
V četrtem poglavju definiramo resonančni graf in pojasnimo povezavo med Kekuléjevimi strukturami in popolnimi prirejanji grafa. Nadalje zapišemo algoritem, ki omogoča iskanje resonančnega grafa poljubnega CERS-a, temelji pa na binarnem kodiranju njegovih popolnih prirejanj. Zatem se ukvarjamo tudi z raziskovanjem CERS-ov, ki imajo izomorfne resonančne grafe. Dobljene rezultate nato uporabimo na fenilenih in tako dobimo zvezo med njihovimi resonančnimi grafi in resonančnimi grafi katakondenziranih benzenoidnih grafov. Na koncu poglavja predstavimo definicijo marjetične kocke in karakteriziramo CERS-e, katerih resonančni grafi so marjetične kocke.
V petem poglavju so predstavljeni topološki indeksi, ki temeljijo na razdaljah v grafu oziroma na stopnjah vozlišč. Nadalje predstavimo krepko utežene grafe in na njih definiramo indekse tipa Szeged.
V zaključku poglavja predstavimo model, s katerim obravnavamo odvisnost med vrelišči alkenov in alkadienov ter povezavno-uteženimi Wienerjevimi indeksi. Pri tem izvedemo nelinearno regresijsko analizo.
V šestem poglavju definiramo kvocientni graf poljubnega povezanega grafa. V nadaljevanju predstavimo posplošeno metodo prerezov in dokažemo, da lahko le-to uporabimo tudi za izračun Schultzevega in Gutmanovega indeksa. Rezultate uporabimo na fenilenih in nekaterih drugih grafovskih družinah. Na koncu šestega poglavja razvijemo posplošeno metodo prerezov za topološke indekse tipa Szeged in zapišemo formulo za izračun teh indeksov za poljuben krepko uteženi graf. Nazadnje ponudimo še nekaj zgledov uporabe izpeljane metode za različne molekularne grafe. Ključne besede: Djoković-Winklerjeva relacija, resonančni graf, benzenoidni sistem, fenilen, CERS, Kekuléjeva struktura, popolno prirejanje, marjetična kocka, kvocientni graf, topološki indeks, Wienerjev indeks, Gutmanov indeks, Schultzev indeks, topološki indeksi tipa Szeged, posplošena metoda prerezov Objavljeno v DKUM: 27.07.2022; Ogledov: 985; Prenosov: 101
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9. Matematika A : za študente FKKT UMPetra Žigert Pleteršek, Matevž Črepnjak, 2022, učbenik za višje in visoke šole Opis: Visokošolski učbenik predstavlja uvod v matematično analizo na naravoslovno-tehniških fakultetah. Poleg osnovnih pojmov o množicah, logiki in številih so obravnavane realne funkcije in z njimi povezana pojma limite funkcije in zveznosti, ter zaporedja in vrste. Zajete se vsebine diferencialnega računa z aplikacijo odvoda, ter integralni račun. Ključne besede: logika, množice, realne funkcije, limita in zveznost funkcije, zaporedja, vrste, odvod funkcije, integral Objavljeno v DKUM: 18.02.2022; Ogledov: 1460; Prenosov: 605
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10. Hereditarnia 2019 : Book of Abstracts, Maribor, 21st & 22nd June, 20192019, druge monografije in druga zaključena dela Opis: The booklet contains the abstracts of the talks given at the 22th Hereditarnia Workshop on Graph Properties that was held at the Faculty of Electrical Engineering and Computer Science in Maribor on 21st and 22nd of June, 2019. The workshop attracted 22 participants from 8 countries. All of the participants are researchers in di˙erent areas of graph theory, but at this event they all presented topics connected with (hereditary) graph properties. Themes of the talks encompass a wide range of contemporary graph theory research, notably, various types of graph colorings, graph domination, some graph dimensions matchings and graph products. Beside the abstracts of the plenary speaker (Roman Sotak) and three invited speakers (Tanja Gologranc, Michael A. Henning and Ismael G. Yero), the booklet also contains the abstracts of 7 contributed talks given at the event. Ključne besede: mathematics, graph theory, Hereditarnia, Maribor, Slovenia Objavljeno v DKUM: 13.12.2019; Ogledov: 1350; Prenosov: 354
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