A characterization of graphs with given maximum degree and smallest possible matching number

In 2018 Henning and Yeo (2018) [9] proved that if G is a connected graph of order n, size m and maximum degree k where k≥3 is odd, then [Formula presented], where α^′(G) is the matching number of G. For k≥3 odd, the k-regular graphs achieving equality in this bound were characterized by O and West (2010) [12]. In this paper we extend this result due to O and West. For k≥3 odd, we characterize all non-regular graphs with maximum degree k that achieve equality in the Henning-Yeo lower bound on the matching number. Together with the O and West characterization of the regular graphs, this provides a complete characterization.