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Opis
Description
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SLO Obstaja več vrst končnih, neusmerjenih, povezanih grafov, ki imajo vrsto kombinatorno simetričnost, ki jo je mogoče analizirati s kombinatoričnimi alialgebrskimi metodami (algebrske metode, kot so linearna algebra, geometrija, meje linearnega programiranja, posebne funkcije, ortogonalnipolinomi, teorija predstavitev in itd.). V tem projektu gradimo in nadaljujemo s preučevanjem $t$-točkovnega števnika, znanstvenega dela, ki smo gazačeli v prispevku ``Enoten pogled na neenakosti za razdaljno-regularne grafe (del I)', J. Combin. Teorija Ser. B 154 (2022) avtorjev ArnoldaNeumaierja in Safeta Penjića. Naslednji korak (v naših mislih) je uporaba števila $t$-točk pri preučevanju odnosov med kombinatorično strukturo inalgebraičnimi lastnostmi grafov. Za to moramo pritegniti druge strokovnjake s tega področja, ki so pripravljeni delati na tej temi. Novi strokovnjakibodo dali drugačen matematični pogled na $t$-točkovno štetje. S tem projektom se nameravamo povezati s strokovnjaki s tega področja z vsaj dvehuniverz (da bi zbrali nove ekipe s teh univerz), da bi delali na `istini` aplikaciji - uporabite štetje $t$ točk iz zgornjega dokumenta na področjuasocijativnih shem. $t$-točkovno štetje iz predlaganega projekta je ``novo`` orodje za obravnavo izračunov z določeno kombinatorično podstrukturo v danem grafu.Podobno kot arabske številke prevzamejo rimske številke, bo naše $t$-točkovno štetje (sčasoma) prevzelo novo orodje pri štetju (za zdaj vkombinatorični teoriji grafov). S tem projektom naš pristop s štetjem $t$-točk ne bo zgolj kombinatoričen: za algebraični pristop je dovolj, da vsakoštevilo na robu zamenjamo z matriko in da dobimo algebraični pristop pri štetju $t$-točk in z njim znotraj sheme povezovanja.
EN There are several types of finite, undirected, connected graphs that have the sort of combinatorial regularity that can be analysed usingcombinatorial or algebraic methods (algebraic methods such as linear algebra, geometry, linear programming bounds, special functions, orthogonalpolynomials, representation theory and etc.). In this project we build up and continue to study $t$-point counts, the scientific work that we started inthe paper ``A unified view of inequalities for distance-regular graphs (part I)'', J. Combin. Theory Ser. B 154 (2022) by Arnold Neumaier and SafetPenjić. Next step (in our mind) is to apply $t$-point counts in study relations between combinatorial structure and algebraic properties of graphs. Todo that we need to bring other experts from the field which are willing to work on the topic. New experts will give different mathematical point ofview to $t$-point counts. With this project we plan to connect with the experts from the field from at least two universities (to pull up new teamsfrom these universities) to work at ``the same'' application - apply $t$-point counts from the above paper in the filed of association schemes. The $t$-point counts from the proposed project are ``new'' tool to deal with computations with certain combinatorial sub-structure in a given graph.Similar as the Arabic numerals take over Roman numerals, our $t$-point counts will (with time) take over as new tool in counting (for now incombinatorial graph theory). With this project our approach with $t$-point counts will not be purely combinatorial: for algebraic approach, it isenough to replace every number on the edge be by a matrix, and to get algebraic approach in $t$-point counts and with it within association scheme. |
