The cardinality of the largest matching in a graph G , denoted by α ′ ( G ) , is referred to as the upper matching number of G . The lower matching number i ′ ( G ) is defined as the cardinality of the smallest maximal matching in G . We introduce the concept of the common matching number of a graph G , denoted by α c ′ ( G ) , which is the largest integer k such that every edge in G belongs to a matching that contains at least k edges. In this paper, we explore the relationships between the parameters i ′ ( G ) , α c ′ ( G ) , and α ′ ( G ) . In particular, we demonstrate that the difference between α c ′ ( G ) and i ′ ( G ) can be arbitrarily large, while the difference between α ′ ( G ) and α c ′ ( G ) can at most be one. Additionally, we characterize the trees T for which i ′ ( T ) = α c ′ ( T ) , as well as the trees T for which α c ′ ( T ) = α ′ ( T ) .

The cardinality of a largest matching of G, denoted by α'(G), is called the upper matching number of G. The lower matching number i'(G) of a graph G is the cardinality of a smallest maximal matching of G. We introduce the concept of the common edge independence number of a graph G, denoted by α'c(G), is the largest integer k such that every edge of G belongs to a matching that has at least k edges. For any graph G, the relations between above parameters are given by the chain of inequalities i'(G) ≤ α'c(G) ≤ α'(G). We study relations between this three parameters, in particular we show that the difference between α'c(G) and i'(G) can be arbitrarily large while α'(G) and α'c(G) may differ by at most one. We also characterize the trees T for which i'(T) = α'c(T), and the trees T for which α'c(T) = α'(T).

A vertex cover of a graph G = ( V, E) is a set X ⊂ V such that each edge of G is incident to at least one vertex of X. The vertex cover number τ( G) is the minimum cardinality of a vertex cover of G. A dominating set D ⊆ V is a weakly connected dominating set of G if the subgraph G[ D] w = ( N[ D], E w ) weakly induced by D, is connected, where E w is the set of all edges having at least one vertex in D. The weakly connected domination number γ w ( G) of G is the minimum cardinality among all weakly connected dominating sets of G. In this article we characterize the graphs where γ w ( G) = τ( G). In particular, we focus our attention on bipartite graphs, regular graphs, unicyclic graphs, block graphs and corona graphs.