Graph Theory With Applications
Graph Theory With Applications
| Hi-index | 5.23 |
One way to generalize the concept of degree in a graph is to consider the neighborhood N(S) of an independent set S instead of a simple vertex. The minimum generalized degree of order t of G is then defined, for 1≤t≤&agr; (the independence number of G), by ut=min{|N(S)|: S (semi-circle right) V; S is independent and |S| = t}. The graph G is said to be u(sub)t-regular if |N(S1)| = |N(S2)| for every pair S1; S2 of independent sets of t elements, totally u(sub)t-regular (resp. totally u(sub)t≤s-regular where s is given ≤&agr;) if it is u(sub)t-regular for every t≤&agr; (resp. for every t≤s), strongly u(sub)t-regular (resp. strongly u(sub)t6s-regular) if |N(S1)| = |N(S2)| for every pair S1; S2 of independent sets of G (resp. every pair of independent sets of order at most s). We determine the strongly u(sub)t≤2-regular graphs and give some properties of the totally u(sub)t≤2-regular and totally u(sub)t-regular graphs. Some of our results improve already known results.