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We prove Gebhard and Sagan's (e)-positivity of the line graphs of tadpoles in noncommuting variables. This implies the e-positivity of these line graphs. We then extend this (e)-positivity result to that of certain cycle-chord graphs, and derive the bivariate generating function of all cyclechord graphs.

We provide a formula for every Schur coefficient in the chromatic symmetric function of a graph in terms of special rim hook tabloids. This formula is useful in confirming the non-Schur positivity of the chromatic symmetric function of a graph, especially when Stanley's stable partition method does not work. As applications, we determine Schur positive fan graphs and Schur positive complete tripartite graphs. We show that any squid graph obtained by adding n leaves to a common vertex on an m-vertex cycle is not Schur positive if m = 2n − 1, and conjecture that neither are the squid graphs with m = 2n − 1.