The e-positivity and Schur positivity of some spiders and broom trees

We obtain the e-positivity or non-e-positivity of some spider graphs with three legs, the positivity classification of all broom graphs, and the positivity classification of most double broom graphs. The methods involve extracting particular e-coefficients of the chromatic symmetric function of these graphs with the aid of Orellana and Scott's triple-deletion property, and using the combinatorial formula of Schur coefficients by examining certain special rim hook tabloids. We conjecture that a spider S(a,b,c) with c≥3 is e-positive if and only if it is S(8,5,3) or S(14,9,5).