Povzetek
A graph ▫$X$▫ is said to be distance-balanced if for any edge ▫$uv$▫ of ▫$X$▫, the number of vertices closer to ▫$u$▫ than to ▫$v$▫ is equal to the number of vertices closer to ▫$v$▫ than to ▫$u$▫. A graph ▫$X$▫ is said to be strongly distance-balanced if for any edge ▫$uv$▫ of ▫$X$▫ and any integer ▫$k$▫, the number of vertices at distance ▫$k$▫ from ▫$u$▫ and at distance ▫$k+1$▫ from ▫$v$▫ is equal to the number of vertices at distance ▫$k+1$▫ from ▫$u$▫ and at distance ▫$k$▫ from ▫$v$▫. Exploring the connection between symmetry properties of graphs and the metric property of being (strongly) distance-balanced is the main theme of this article. That a vertex-transitive graph is necessarily strongly distance-balanced and thus also distance-balanced is an easy observation. With only a slight relaxation of the transitivity condition, the situation changes drastically: there are infinite families of semisymmetric graphs (that is, graphs which are edge-transitive, but not vertex-transitive) which are distance-balanced, but there are also infinite families of semisymmetric graphs which are not distance-balanced. Results on the distance-balanced property in product graphs prove helpful in obtaining these constructions. Finally, a complete classification of strongly distance-balanced graphs is given for the following infinite families of generalized Petersen graphs: GP▫$(n,2)$▫, GP▫$(5k+1,k)$▫, GP▫$(3k 3,k)$▫, and GP▫$(2k+2,k)$▫.
Ključne besede
graph theory;graph;distance-balanced graphs;vertex-transitive;semysimmetric;generalized Petersen graph;
Podatki
Jezik: |
Angleški jezik |
Leto izida: |
2006 |
Tipologija: |
1.01 - Izvirni znanstveni članek |
Organizacija: |
UP - Univerza na Primorskem |
UDK: |
519.17 |
COBISS: |
1714391
|
ISSN: |
0012-365X |
Št. ogledov: |
4078 |
Št. prenosov: |
89 |
Ocena: |
0 (0 glasov) |
Metapodatki: |
|
Ostali podatki
Sekundarni jezik: |
Angleški jezik |
Sekundarne ključne besede: |
teorija grafov;razdaljno uravnoteženi grafi;polsimetrični grafi;posplošeni Petersenov graf;tranzitivni grafi; |
Vrsta dela (COBISS): |
Delo ni kategorizirano |
Strani: |
str. 1881-1894 |
Letnik: |
ǂVol. ǂ306 |
Zvezek: |
ǂiss. ǂ16 |
Čas izdaje: |
2006 |
ID: |
1471767 |