# American Institute of Mathematical Sciences

February  2009, 3(1): 97-114. doi: 10.3934/amc.2009.3.97

## Edge number, minimum degree, maximum independent set, radius and diameter in twin-free graphs

 1 Institut TELECOM - TELECOM ParisTech, & Centre National de la Recherche Scientifique - LTCI UMR 5141, 46, rue Barrault, 75634 Paris Cedex 13, France 2 Department of Mathematics, University of Turku, 20014 Turku, Finland

Received  November 2008 Revised  January 2009 Published  January 2009

Consider a connected, undirected graph $G=(V,E)$ and an integer $r \geq 1$; for any vertex $v\in V$, let $B_r(v)$ denote the ball of radius $r$ centred at $v$, i.e., the set of all vertices linked to $v$ by a path consisting of at most $r$ edges. If for all vertices $v \in V$, the sets $B_r(v)$ are different, then we say that $G$ is $r$-twin-free.
In $r$-twin-free graphs, we prolong the study of the extremal values that can be achieved by the main classical parameters in graph theory, and investigate here the number of edges, the minimum degree, the size of a maximum independent set, as well as radius and diameter.
Citation: David Auger, Irène Charon, Iiro Honkala, Olivier Hudry, Antoine Lobstein. Edge number, minimum degree, maximum independent set, radius and diameter in twin-free graphs. Advances in Mathematics of Communications, 2009, 3 (1) : 97-114. doi: 10.3934/amc.2009.3.97
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