On The Minimum Average Lower Independence Number In Unicyclic And Bicyclic Graphs

For a vertex v of a graph G = (V, E), the average lower independence number i_av(G) of G is defined as1/|V|Σ_v(?)Vi_v(G) , where i_v(G) is the minimum cardinality of a maximal independent set that contains v. i(G) of G is defined as the minimum cardinality of a maximal independent set of G. 7(G) is the minimum cardinality of dominating set of G. For any vertex v (?) V(G), the matching number β(G) of G is the cardinality of a largest matching in G and β_v{G) is the maximum cardinality of a matching in G — N[v], that is β_v(G) = β(G — N[v]).Henning (Trees with equal average domination and independent domination numbers, Ars Commbin. 71 (2004) 305-318) showed that for a tree T with order n ≥ 2, i_av(T) ≤n — 2 +(2/n) Blidia et al. (On average lower independence and domination numbers in graphs, Discrete Mathematics 295 (2005) 1-11) showed that any graph G with n vertices and m edges has i_av{G) ≤n —2m/n—(1/n)Σ_v(?)V(G)β_v(G)Inspired by those results, we determine the upper bound of the average lower independence number of unicyclic graphs and bicyclic graphs. Our main results are as follows:(1) Let G be a unicyclic graph with order n ≥ 5. Theni_av(G)≤n-3 +(3/n) , with equality if and only if G is the graph obtained from the star S_n on n vertices by joining its two vertices of degree one.(2) Let G be a bicyclic graph with order n ≥6. Then2i_av(G)