The chromaticity of the join of tree and null graph
The chromaticity of the graph <i>G</i>, which is join of the tree T<sub>p</sub> and the null graph O<sub>q</sub>, is studied. We prove that <i>G</i> is chromatically unique if and only if 1 ≤ p ≤ 3, 1 ≤ q ≤ 2; a graph <i>H</i> and T<sub>p</sub> + O<sub>p-1</sub> are χ-equivalent if and only if H = T′<sub>p</sub> + O<sub>p-1</sub>, where T′<sub>p</sub> is a tree of order p; H and T<sub>p</sub> + O<sub>p</sub> are χ-equivalent if and only if H ∈ {T′<sub>p</sub> + O<sub>p</sub>, T″<sub>p+1</sub> + O<sub>p-1</sub>}, where T′<sub>p</sub> is a tree of order p, T″<sub>p+1</sub> is a tree of order p + 1. We also prove that if p ≤ q, then χ′(G) = ch′(G) = ∆(G); if ∆(G) = |V(G)| - 1, then χ′(G) = ch′(G) = ∆(G) if and only if G ≠ K<sub>3</sub>.