Boštjan Brešar (Author), Sandi Klavžar (Author), Riste Škrekovski (Author)

Abstract

Naj bo ▫$\alpha_i(G)$▫ število induciranih ▫$i$▫-kock grafa ▫$G$▫. Tedaj je polinom kock ▫$c(G,x)$▫ grafa ▫$G$▫ definiran z ▫$\sum_{i \ge 0} \alpha_i (G) x_i$▫. Pokazano je, da je vsaka funkcija ▫$f$▫ z dvemi predpisanimi naravnimi lastnostmi do faktorja ▫$f(Q_0,x)$▫ enaka polinomu kock. Vpeljan je tudi odvod ▫$\partial G$▫ medianskega grafa ▫$G$▫. Dokazano je, da je polinom kock edina funkcija ▫$f$▫ z lastnostjo ▫$f'(G,z) = f(\partial G,x)$▫, če je le ▫$f(G,0) = |V(G)|$▫. Dokazanih je tudi več relacij za medianske grafe, ki posplošujejo prej znane rezultate. Na primer, za vsak ▫$s \ge 0$▫ velja ▫$c^{(s)}(G, x+1) = \sum_{i \ge s} \frac{c^{(s)}(G,x)}{(i-s)!}$▫.

Keywords

matematika;teorija grafov;polinom kock;odvod grafa;medianski grafi;mathematics;graph theory;cube polynomials;graph derivation;median graphs;

Data

Language: English
Year of publishing:
Typology: 1.01 - Original Scientific Article
Organization: UL FMF - Faculty of Mathematics and Physics
UDC: 519.17
COBISS: 12165977 Link will open in a new window
ISSN: 1077-8926
Views: 30
Downloads: 6
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Other data

Secondary language: Slovenian
Secondary title: Polinom kock in njegovi odvodi: primer medianskih grafov
Secondary abstract: For ▫$i \ge 0$▫, the ▫$i$▫-cube of ▫$Q_i$▫ is the graph on ▫$2^i$▫ vertices representing ▫$0/1$▫ tuples of lenght ▫$i$▫, where two vertices are adjacent whenever the tuples differ in exactly one position. (In particular, ▫$Q_0=K_1$▫.) Let ▫$\alpha_i(G)$▫ be the number of induced ▫$i$▫-cubes of a graph ▫$G$▫. Then the cube polynomial ▫$c(G,x)$▫ of ▫$G$▫ is introduced as ▫$\sum_{i \ge 0} \alpha_i (G) x_i$▫. It is shown that any function ▫$f$▫ with two related, natural properties, is up to the factor ▫$f(Q_0,x)$▫ the cubes polynomial. The derivation ▫$\partial G$▫ of a median graph ▫$G$▫ is also introduced and it is proved that the cubes polynomial is the only function ▫$f$▫ with the property ▫$f'(G,z) = f(\partial G,x)$▫ provided that ▫$f(G,0) = |V(G)|$▫. As the main application of the new concept,several relations that widely generalize previous such results for median graphs are proved. For istance, it is shown that for any ▫$s \ge 0$▫ we have ▫$c^{(s)}(G, x+1) = \sum_{i \ge s} \frac{c^{(i)}(G,x)}{(i-s)!}$▫, where certain derivatives of the cube polynomial coincide with well-known invariants of median graphs.
Secondary keywords: matematika;teorija grafov;polinom kock;odvod grafa;medianski grafi;
URN: URN:SI:UM:
Type (COBISS): Not categorized
Pages: R3 (11 str.)
Volume: ǂVol. ǂ10
Issue: ǂno. ǂ1
Chronology: 2003
ID: 1472232