The Hosoya-Wiener Polynomial of Weighted Trees

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

Computing the Weighted Wiener and Szeged Number on Weighted Cactus Graphs in Linear Time

Cactus is a graph in which every edge lies on at most one cycle. Linear algorithms for computing the weighted Wiener and Szeged numbers on weighted cactus graphs are given. Graphs with weighted vertices and edges correspond to molecular graphs with heteroatoms

Education Equity in Times of Emergency Remote Teaching: The Case of Slovenia

During the COVID -19 situation, it was often warned that emergency remote teaching increases differences among students. Additionally, some empirical results in Slovenia indicate that the situation at schools in Slovenia was very diverse, leading to a violation of the equity principle in education. In this paper, we investigate teaching methods used by teachers in crisis teaching. The database presents 61 diaries of future teachers from the first grade of elementary school to the fourth grade of secondary school. The results show differences between mentor-teachers emergency remote teaching strategies. Differences are also statistically significant according to the educational stages. The results suggest that secondary school teachers have most effectively adopted and integrated different distance learning strategies into their work. On the other hand, some class teachers have not been as successful, probably also due to the students’ distinctive characteristics. The primary purpose of this paper is to describe the Razlagamo.si learning environment, which can reduce differences by providing a common educational point for all Slovenian primary and secondary school students. Finally, we give some implications for physical re-engagement at school

Izračun uteženega Wienerjevega in Szegedovega števila uteženih kaktusov v linearnem času

Cactus is a graph in which every edge lies on at most one cycle. Linear algorithms for computing the weighted Wiener and Szeged numbers on weighted cactus graphs are given. Graphs with weighted vertices and edges correspond to molecular graphs with heteroatoms.Kaktus je graf, kjer vsaka povezava leži največ na enam ciklu. Podan je linearni algoritem za izračun uteženega Wienerjevega in Szededovega števila uteženih kaktusov

Hosoya-Wienerov polinom za utežene grafove

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.Formule za Wienerov broj i Hosoya-Wienerov polinom grafova s uteženim čvorovima i granama izražene su pomoću doprinosa grana i puteva. U slučaju stabala s korijenom Hosoya-Wienerov polinom izražen je kao zbroj doprinosa čvorova. Dana je također rekurzivna formula za račun Hosoya-Wienerovog polinoma uteženih stabala

Prepoznavanje uteženih usmerjenih kartezičnih grafovskih svežnjev

In this paper we show that methods for recognizing Cartesian graph bundles can be generalized to weighted digraphs. The main result is an algorithm which lists the sets of degenerate arcs for all representations of digraph as a weighted directed Cartesian graph bundle over simple base digraphs not containing transitive tournament on three vertices. Two main notions are used.The first one is the new relation ▫$vec{delta}^ast$▫ defined among the arcs of a digraph as a weighted directed analogue of the well-known relation ▫$delta^ast$▫. The second one is the concept of half-convex subgraphs. A subgraph ▫$H$▫ is half-convex in ▫$G$▫ if any vertex ▫$x in G setminus H$▫ has at most one predecessor and at most one successorPredstavljena je posplošitev prepoznavanja kartezičnih grafovskih svežnjev za utežene usmerjene grafe. Osrednji rezultat predstavlja algoritem, ki vrne množice degeneriranih vektorjev vseh predstavitev usmerjenih grafov v obliki uteženih usmerjenih kartezičnih grafovskih svežnjev nad baznimi grafi brez tranzitivnih turnirjev na treh točkah. Temeljna pojma pri izpeljavi tega rezutata sta relacija ▫$vec{delta}^ast$▫ in polkonveksnost. Relacija ▫$vec{delta}^ast$▫, definirana na množici vektorjev usmerjenega grafa, predstavlja posplošitev znane relacije ▫$delta^ast$▫. Podgraf ▫$H$▫ je polkonveksen v ▫$G$▫, če ima poljubna točka ▫$x in G setminus H$▫ največ enega predhodnika in največ enega naslednika

On recognizing Cartesian graph bundles

AbstractGraph bundles generalize the notion of covering graphs and graph products. In Imrich et al. (Discrete Math. 167/168 (1988) 393–403.) an algorithm that finds a presentation as a nontrivial Cartesian graph bundle for all graphs that are Cartesian graph bundles over triangle-free simple base was given. In this paper we extend this algorithm to recognize Cartesian graph bundles over a K4⧹e-free simple base, without induced K3,3. Finally, we conjecture the existence of algorithm for recognition of Cartesian graph bundle over a K4⧹e-free simple base

Interactivity and e-learning materials E-um

Ena od pomembnih prednosti e-učnih gradiv v primerjavi s tradicionalnimi tiskanimi učnimi gradivi je interaktivnost tega medija, ki omogoča interakcijo med učencem in učno vsebino v e-učnem gradivu. Po pregledu znanstvene in strokovne literature smo ugotovili, da ni enotne opredelitve interaktivnosti, zato so nas v prispevku zanimale skupne značilnosti različnih opredelitev interaktivnosti in na podlagi tega smo nato predstavili tudi lastno opredelitev tega pojma. Na sprejetih teoretičnih izhodiščih v nadaljevanju so opredeljeni parametri za določanje stopenj interaktivnosti, splošna opredelitev interaktivnosti pa je aplicirana na opis interaktivnosti učnih medijev. Ob sklepu je podana analiza interaktivnosti gradnikov v e-učnih gradivih spletnega učnega portala E-um.One of the main advantages of e-learning materials in comparison with traditional printed materials is interactivity of media that permits high level of interaction between students and learning objects. It is interesting that in scientific literature there is still no common agreement about definition of interactivity. Therefore in this article we present some connections between those different premises on interactivity and we introduce our own definition of interactivity in communication process. On that base we ground the parameters for describing levels of interactivity and we apply the general definition of interactivity to definition of interactivity of learning objects. Finally we analyse the interactivity of e-devices in program eXe E-um which was developed to create e-learning materials for project E-um

Problem nezaželenega centra na obteženih kaktus grafih

Problem nezaželenih centrov v grafu predstavlja določitev takšne lokacije na povezavah grafa, da je njena minimalna razdalja do poljubne točke grafa kolikor se da velika. Uteži na točkah grafa lahko predstavljajo njihovo občutljivost, ki jo je moč oceniti z eno izmed konstantno mnogo vrednosti. Kadar je vsaki točki grafa prirejena ena izmed ▫$c$▫ različnih vrednosti (uteži) glede na njeno občutljivost, rešujemo tako imenovan problem nezaželenih centrov na grafu z ovrednotenimi točkami. V tem članku bomo predstavili algoritem, ki določi nezaželeni center na kaktusu z ovrednotenimi točkami v linearnem času ▫$O(cn)$▫, kjer je ▫$n$▫ število točk in ▫$c$▫ število uteži.The obnoxious center problem in a graph ▫$G$▫ asks for a location on an edge of the graph such that the minimum weighted distance from this point to a vertex of the graph is as large as possible. An algorithm is given which finds the obnoxious center on a weighted cactus graph in ▫$O(cn)$▫ time, where ▫$n$▫ is the number of vertices and ▫$c$▫ is the number of different vertex weights (called marks)