In this paper, we proceed on determining the minimum $q_{ min}$ among the connected nonbipartite graphs on $n\geq 5$ vertices and with domination number $\frac{ n+1}{ 3}<\gamma\leq \frac{ n-1}{ 2}$. Further results obtained are as follows: $\mathrm{ (i)}$ among all nonbipartite connected graph of order $n\geq 5$ and with domination number $\frac{ n-1}{ 2}$, the minimum $q_{ min}$ is completely determined; $\mathrm{ (ii)}$ among all nonbipartite graphs of order $n\geq 5$, with odd-girth $g_{ o}\leq5$ and domination number at least $\frac{ n+1}{ 3}<\gamma\leq \frac{ n-2}{ 2}$, the minimum $q_{ min}$ is completely determined.
Categories: math.CO