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

Abstract

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 ) .