University of Primorska
Faculty of Mathematics, Natural Sciences and Information Technologies
Več informacij o projektu / More info about the project
Opis / Description
SLO:
Naj bo G dvodelni graf z množico vozlišč Y È Y' in biparticijo (Y,Y’). Predpostavljamo, da ima vsako vozlišče iz množice Y ekscentričnost D. Izberemotri vozlišča: vozlišče x iz množice Y, vozlišče y, ki je na razdalji 2 od vozlišča x, in vozlišče z, ki je na razdalji i tako od x kot tudi od y. Za vsako celoštevilo i (1 leq i leq D) naj bo $gamma_i(x, y, z)$ število skupnih sosedov vozlišč x in y, ki so na razdalji i-1 od vozlišča z. Osrednji cilj tega projektaje identifikacija lastnosti dvodelnih grafov G, ki imajo to lastnost, da je število $gamma_i(x, y, z)$ neodvisno od izbire vozlišč x, y, z. Če to velja zavsa cela števila $i ; (1 leq i leq D-2)$, potem rečemo, da je graf G skoraj 2-Y-homogen. Če ta pogoj velja tudi, ko je i = D - 1, potem rečemo, da jegraf G 2-Y-homogen.
Karakterizacija oziroma sistematična klasifikacija dvodelnih grafov, ki so (skoraj) 2-Y-homogeni, predstavlja izjemno zahteven izziv. V splošnem seje ta problem pokazal kot izjemno težek in kompleksen, zato se večina raziskav osredotoča na specifične družine grafov, in sicer na dvodelnerazdaljno-regularne grafe in razdaljno-biregularne grafe.
Naš pristop temelji na obstoječi literaturi in obravnava izzive v okviru razdaljno-polregularnih grafov. Namen projekta je raziskovanje 2-Y-homogenihlastnosti razdaljno-polregularnih grafov in klasifikacija vseh 2-Y-homogenih razdaljno-biregularnih grafov. Poleg glavnih ciljev ima naša raziskava potencial za odmevne rezultate, ki bodo segali izven okvirov stroke ter imeli pomemben vpliv tudi na širšo skupnost. Rezultati raziskave ne bodo uporabni samo v teoriji grafov, ampak tudi na ostalih področjih matematike, kot sta algebraična kombinatorika in teorija kombinatoričnih načrtov.
NAČRT ZA RAVNANJE Z RAZISKOVALNIMI PODATKI projekta Z1-70008
EN:
Let $Gamma$ be a bipartite graph with vertex set $Y cup Y'$ and bipartition $(Y,Y')$. Assume that each vertex in the set $Y$ has eccentricity $D$. Choose three vertices: vertex $x$ from the set $Y$, vertex $y$ at distance $2$ from vertex $x$, and vertex $z$ at distance $i$ from both $x$ and$y$. For each integer $i$ $(1 leq i leq D)$, let $gamma_i(x, y, z)$ denote the number of common neighbors of vertices $x$ and $y$ at distance $i-1$ from vertex $z$. The central aim of this project is to identify properties of bipartite graphs $Gamma$ such that the quantity $gamma_i(x, y, z)$is independent of the choice of vertices $x$, $y$, and $z$. If this holds for all integers $i$ $(1 leq i leq D-2)$, then we say that the graph$Gamma$ is almost $2$-$Y$-homogeneous. If this condition also holds when $i = D - 1$, then we say that the graph $Gamma$ is $2$-$Y$-homogeneous.
The characterization or systematic classification of bipartite graphs that are (almost) $2$-$Y$-homogeneous presents an extremely challenging task. In general, this problem has proven to be highly difficult and complex, so most research focuses on specific families of graphs, particularly on bipartite distance-regular graphs and distance-biregular graphs.
The characterization or systematic classification of bipartite graphs that are (almost) $2$-$Y$-homogeneous presents an extremely challenging task. In general, this problem has proven to be highly difficult and complex, so most research focuses on specific families of graphs, particularly on bipartite distance-regular graphs and distance-biregular graphs.
Our approach is based on existing literature and addresses challenges within the framework of distance-polynomial regular graphs. The purpose of the project is to investigate $2$-$Y$-homogeneous properties of distance-polynomial regular graphs and classify all $2$-$Y$-homogeneous distance-biregular graphs. In addition to the main objectives, our research has the potential to yield significant results extending beyond the realm of the discipline and impacting the broader community. The outcomes of the research will not only be applicable in graph theory but also in other areas of mathematics such as algebraic combinatorics and combinatorial designs.