The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two - Bai Yongqiang; Ma Hongping; Zhang Xia (Open Mathematics)
The number of jj-matchings in a graph HH is denote by m(H,j)m\left(H,j). If for two graphs H1H_{1} and H2H_{2}, m(H1,j)≥m(H2,j)m\left(H_{1},j)\ge m\left(H_{2},j) for all jj, then we write H1≽H2H_{1}\succcurlyeq {H}_{2}. If H1≽H2H_{1}\succcurlyeq {H}_{2}, and m(H1,i)>m(H2,i)m\left(H_{1},i)\gt m\left(H_{2},i) for some ii, then we write H1≻H2H_{1}\hspace{0.33em}\succ \hspace{0.33em}{H}_{2}. In this article, by utilizing several new graph transformations, we determine the least element with respect to the quasi-order ≽\succcurlyeq among all unicyclic graphs with fixed order and number of vertices of degree two. As consequences, we characterize the graphs with minimum matching energy and with minimum Hosoya index in the set of all unicyclic graphs with fixed order and number of vertices of degree two.