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1.
On median and quartile sets of ordered random variables
Iztok Banič, Janez Žerovnik, 2020, original scientific article

Abstract: We give new results about the set of all medians, the set of all first quartiles and the set of all third quartiles of a finite dataset. We also give new and interesting results about rela- tionships between these sets. We also use these results to provide an elementary correctness proof of the Langford's doubling method.
Keywords: statistic, probability, median, first quartile, third quartile, median set, first quartile set, third quartile set
Published in DKUM: 19.09.2022; Views: 97; Downloads: 5
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2.
More results on the domination number of Cartesian product of two directed cycles
Ansheng Ye, Fang Miao, Zehui Shao, Jia-Bao Liu, Janez Žerovnik, Polona Repolusk, 2019, original scientific article

Abstract: Let γ(D) denote the domination number of a digraph D and let C$_m$□C$_n$ denote the Cartesian product of C$_m$ and C$_n$, the directed cycles of length n ≥ m ≥ 3. Liu et al. obtained the exact values of γ(C$_m$□C$_n$) for m up to 6 [Domination number of Cartesian products of directed cycles, Inform. Process. Lett. 111 (2010) 36–39]. Shao et al. determined the exact values of γ(C$_m$□C$_n$) for m = 6, 7 [On the domination number of Cartesian product of two directed cycles, Journal of Applied Mathematics, Volume 2013, Article ID 619695]. Mollard obtained the exact values of γ(C$_m$□C$_n$) for m = 3k + 2 [M. Mollard, On domination of Cartesian product of directed cycles: Results for certain equivalence classes of lengths, Discuss. Math. Graph Theory 33(2) (2013) 387–394.]. In this paper, we extend the current known results on C$_m$□C$_n$ with m up to 21. Moreover, the exact values of γ(C$_n$□C$_n$) with n up to 31 are determined.
Keywords: domination number, Cartesian product, directed cycle
Published in DKUM: 02.09.2022; Views: 145; Downloads: 4
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3.
Simetrijske grupe končnih vzorcev
Matej Mencinger, 2021

Abstract: Končni vzorci so najprej definirani intuitivno, kasneje pa še eksaktno (matematično). Simetrije končnih vzorcev so definirane s pomočjo izometrij ravnine. Obravnavani so štirje osnovni razredi izometrij ravnine: zrcaljenje, rotacija, translacija ter drsno zrcaljenje. V klasifikacijskem izreku je dokazano, da vsaka izometrija spada v enega od osnovnih štirih razredov. Pri obravnavi vektorjev in matrik se omejimo na ravnino in trirazsežni vektorski prostor. Dokazano je, da izometrijam ravnine s fiksno točko pripadajo natanko ortogonalne matrike. V poglavju o grupah so obravnavani pojmi: (pod)grupa, red grupe, izomorfizem grup ter generatorji grupe. Glavni rezultat je klasifikacija simetrijskih grup končnih vzorcev v ciklične ali diedrske, kar danes imenujemo Leonardov izrek. Podano je tudi nekaj informacij o zgodovinski dobi, v kateri je deloval Leonardo da Vinci in nekaterih povezavah med njegovim delom in matematiko. Učbenik je namenjen študentom arhitekture in vsebuje številne primere, rešene naloge ter obsežno slikovno gradivo.
Keywords: izometrija ravnine, končni vzorec, simetrija, končna grupa, Leonardov izrek
Published in DKUM: 21.12.2021; Views: 580; Downloads: 118
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4.
Koordinacija blagovnega toka v maloprodajni oskrbovalni verigi
Tea Vizinger, 2019, doctoral dissertation

Abstract: Disertacija obravnava koordinacijo blagovnih tokov v maloprodajnih oziroma trgovinskih oskrbovalnih verigah. Zanje je značilno, da delujejo v nenehno spreminjajočem se okolju ter so močno odvisne od povpraševanja končnih odjemalcev. Tipično maloprodajno oskrbovalno verigo sestavlja več trgovin, ki so oskrbovane iz vsaj enega skladišča izbranega trgovca. Da dosežemo usklajenost zalog med objekti dane oskrbovalne verige, je zagotovo najprej potrebno doseči primerno koordinacijo vseh povezanih in odvisnih si aktivnosti, ki spremljajo blagovni tok. Pri gradnji matematičnega modela smo se najprej osredotočili na analizo povpraševanja, medtem ko so dinamično prilagajanje cen, upoštevanje pokvarljivosti itd. predmet nadaljnih raziskav, povezanih s komplementarnim modelom za operativno planiranje blagovnega toka. Idejno je pristop zasnovan tako, da se najprej poišče primerni distribucijski plan (taktični plan) z minimalnimi pričakovanimi stroški distribucije, kot tudi z minimalnimi pričakovanimi stroški rizikov. Problem predstavljamo kot večkriterijski problem optimizacije, predlagani stohastični model za dosego koordinacije pa pri tem kombinira več metod in postopkov iz področij teorije grafov, teorije verjetnosti in operacijskih raziskav. Ker je izbrani problem računsko kompleksen, za njegovo reševanje uporabimo nekaj hevrističnih postopkov lokalnega iskanja ter tudi predlagamo robustni optimizacijski pristop reševanja danega problema. Eksperimentalna študija pokaže, da je za optimizacijski problem mogoče implementirati algoritem, ki da uporabne rešitve.
Keywords: Distribucija, Zaloge, Optimizacija, Hevristike, Stohastično programiranje
Published in DKUM: 01.10.2019; Views: 1319; Downloads: 113
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5.
Heuristics for NP-hard optimization problems : simpler is better !?
Janez Žerovnik, 2015, original scientific article

Abstract: We provide several examples showing that local search, the most basic metaheuristics, may be a very competitive choice for solving computationally hard optimization problems. In addition, generation of starting solutions by greedy heuristics should be at least considered as one of very natural possibilities. In this critical survey, selected examples discussed include the traveling salesman, the resource-constrained project scheduling, the channel assignment, and computation of bounds for the Shannon capacity.
Keywords: optimization, metaheuristics, local search, greedy construction, traveling salesman problem
Published in DKUM: 17.11.2017; Views: 1211; Downloads: 219
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6.
Roman domination number of the Cartesian products of paths and cycles
Polona Repolusk, Janez Žerovnik, 2012, original scientific article

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: 1055; Downloads: 212
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7.
Altered Wiener indices
Damir Vukičević, Janez Žerovnik, 2005, original scientific article

Abstract: Recently Nikolić, Trinajstić and Randić put forward a novel modification ▫$^mW(G)$▫ of the Wiener number ▫$W(G)$▫, called modified Wiener index, which definition was generalized later by Gutman and the present authors. Here we study another class of modified indices defined as ▫$W_{min,λ}(G) = ∑(V(G)^λm_G(u,ν)^λ−m_G(u,ν)^{2λ})$▫ and show that some of the important properties of ▫$W(G)$▫, ▫$^mW(G)$▫ and ▫$^λW(G)$▫ are also properties of ▫$W_{min,λ}(G)$▫, valid for most values of the parameter λ. In particular, if ▫$T_n$▫ is any n-vertex tree, different from the n-vertex path ▫$P_n$▫ and the n-vertex star ▫$S_n$▫, then for any λ ≥ 1 or λ < 0, ▫$^W_{min,λ}(P_n) > W_{min,λ}(T_n)>W_{min,λ}(S_n)$▫. Thus for these values of the parameter λ, ▫$W_{min,λ}(G)$▫ provides a novel class of structure-descriptors, suitable for modeling branching-dependent properties of organic compounds, applicable in QSPR and QSAR studies. We also demonstrate that if trees are ordered with regard to ▫$W_{min,λ}(G)$▫ then, in the general case, this ordering is different for different λ.
Keywords: mathematics, chemical graph theory, Wiener index, modified Wiener index
Published in DKUM: 17.08.2017; Views: 803; Downloads: 106
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8.
On vulnerability measures of networks
Rija Erveš, Darja Rupnik Poklukar, Janez Žerovnik, 2013, published scientific conference contribution

Abstract: As links and nodes of interconnection networks are exposed to failures, one of the most important features of a practical networks design is fault tolerance. Vulnerability measures of communication networks are discussed including the connectivities, fault diameters, and measures based on Hosoya- Wiener polynomial. An upper bound for the edge fault diameter of product graphs is proved.
Keywords: discrete optimization, communication network, vulnerability
Published in DKUM: 21.07.2017; Views: 788; Downloads: 100
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9.
The Hosoya-Wiener polynomial of weighted trees
Blaž Zmazek, Janez Žerovnik, 2007, original scientific article

Abstract: Formulas for the Wiener number and the Hosoya-Wiener polynomial of edge and vertex weighted graphs are given in terms of edge and path contributions. For a rooted tree, the Hosoya-Wiener polynomial is expressed as a sum of vertex contributions. Finally, a recursive formula for computing the Hosoya-Wiener polynomial of a weighted tree is given.
Keywords: mathematics, graph theory, Hosoya-Wiener polynomial, weighted tree, vertex weighted graphs
Published in DKUM: 05.07.2017; Views: 887; Downloads: 93
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10.
Simplified computation of matchings in polygraphs
Ante Graovac, Damir Vukičević, Damir Ježek, Janez Žerovnik, 2005, original scientific article

Abstract: Matching polynomial and perfect matchings for fasciagraphs, rotagraphs and twisted rotagraphs are treated in the paper. Classical transfer matrix approach makes it possible to get recursions for matching polynomial and perfect matchings, but the order of the matrix grows exponentially in the number of the linking edges between monographs. Novel transfer matrices are introduced whose order is much lower than that in classical transfer matrices. The virtue of the method introduced is especially pronounced when twoor more linking edges end in the same terminal vertex of a monograph. An example of a polyacene polygraph with extended pairings is given where a novel matrix has only 16 entries as compared to 65536 entries in the classical transfer matrix. However, all pairings are treated here on equal footing, but the method introduced can be applied to selected types of pairings of interest in chemistry.
Keywords: polygraphs, matching polynomial, matchings, perfect matchings, Kekulé structures, extended structures, recursive enumeration, transfer matrix method
Published in DKUM: 05.07.2017; Views: 1083; Downloads: 80
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