One way to generalize the concept of degree in a graph is to consider the n
eighborhood N(S) of an independent set S instead of a simple vertex. The mi
nimum generalized degree of order t of G is then defined, for I less than o
r equal to t less than or equal to alpha (the independence number of G), by
u(t) = min {\N(S)\: S subset of V, S is independent and \S \ = t}. The gra
ph G is said to be u(t)-regular if \N(S-t)\ = \N(S-2)\ for every pair S-1,S
-2 of independent sets of t elements, totally u(t)-regular (resp. totally u
(t less than or equal tos)-regular where s is given less than or equal to a
lpha) if it is u(t)-regular for every t less than or equal to alpha (resp.
for every t less than or equal to s), strongly u(t)-regular (resp. strongly
u(t less than or equal tos)-regular) if \N(S-1)\ = \N(S-2)\ for every pair
S-1,S-2 of independent sets of G (resp. every pair of independent sets of
order at most s). We determine the strongly u(t less than or equal to2)-reg
ular graphs and give some properties of the totally u(t less than or equal
to2)-regular and totally u(t)-regular graphs. Some of our results improve a
lready known results. (C) 2001 Elsevier Science B.V. All rights reserved.