Inequalities Connecting the Annihilation and Independence Numbers
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Given a graph G, the number of its vertices is represented by n(G), while the number of its edges is denoted as m(G). An independent set in a graph is a set of vertices where no two vertices are adjacent to each other and the size of the maximum independent set is denoted by α(G). A matching in a graph refers to a set of edges where no two edges share a common vertex and the maximum matching size is denoted by μ(G). If α(G) + μ(G) = n(G), then the graph G is called a König-Egerváry graph. Considering a graph G with a degree sequence d_1 ≤ d_2 ≤⋯≤ d_n, the annihilation number a(G) is defined as the largest integer k such that the sum of the first k degrees in the sequence is less than or equal to m(G) (Pepper, 2004). It is a known fact that α(G) is less than or equal to a(G) for any graph G. Our goal is to estimate the difference between these two parameters. Specifically, we prove a series of inequalities, including a(G) - α(G) ≤μ(G) - 1/2 for trees, a(G) - α(G) ≤ 2 + μ(G) - 2√(1 + μ(G)) for bipartite graphs and a(G) - α(G) ≤μ(G) - 2 for König-Egerváry graphs. Furthermore, we demonstrate that these inequalities serve as tight upper bounds for the difference between the annihilation and independence numbers, regardless of the assigned value for μ(G).
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