Lower bounds of graph energy in terms of matching number

The energy E(G) of a graph G is the sum of the absolute values of all eigenvalues of G. We are interested in the relation between the energy of a graph G and the matching number μ(G) of G. It is proved that E(G)≥2μ(G) for every graph G, and E(G)≥2μ(G)+55c1(G) if the cycles of G (if any) are pairwise vertex-disjoint, where c1(G) denotes the number of odd cycles in G. Besides, we prove that E(G)≥r(G)+12 if G has at least one odd cycle and it is not of full rank, where r(G) is the rank of G.