Two cycle-chord graphs are $e$-positive

TL;DR

This paper proves the $e$-positivity of line graphs of tadpoles and certain cycle-chord graphs, extending known results and providing a bivariate generating function for these graphs.

Contribution

It extends the $e$-positivity proof to cycle-chord graphs and derives their bivariate generating function, building on Gebhard and Sagan's work.

Findings

Proves $e$-positivity of line graphs of tadpoles.

Extends $e$-positivity to certain cycle-chord graphs.

Derives bivariate generating function for cycle-chord graphs.

Abstract

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 cycle-chord graphs.