The matching energy of a graph

The bipartite unicyclic graphs with the first⌊n−34⌋largest matching energies

“…Lemma 6 ( [22,28]). If F is a forest with n(n ≥ 6) vertices, then F P n , with F ∼ P n if and only if F ∼ = P n .…”

“…In [28], the authors gave some elementary results on the matching energy and obtained that ME(S + n ) ≤ ME(G) ≤ ME(C n ) for any unicyclic graph G of order n, where S + n is the graph obtained by adding a new edge to the star S n . In [37], Ji et al characterized the graphs with the extremal matching energy among all bicyclic graphs, while Chen and Shi [10] proved the same extremal results for tricyclic graphs.…”

“…Especially, we call the graphs in B n,n as bipartite unicyclic graphs, which is an interesting class of graphs. What's more, we introduce some new notations appearing in [28] and [40]. The sun graph, denoted by C l (P s 1 +1 , .…”

“…Similar to graph energy, in [28], Gutman and Wagner proposed the concept of matching energy. They defined the matching energy of a graph G as…”

“…In [28], Gutman and Wagner pointed out that the matching energy is a quantity of relevance for chemical applications. They arrived at the simple relation:…”