arXiv subjects

Aarush Vailaya

Publications and source records attributed to Aarush Vailaya.

3 recordsLinked to original sources

arXiv2026

The Tutte symmetric matrix of a graph

We provide a matrix-based formula for the Tutte symmetric function of a graph. In particular, for any graph $G$ with a designated head and tail vertex, we describe an infinite matrix $M_G$ from which the Tutte symmetric function can be easily recovered. We prove gluing graphs together corresponds to matrix multiplication, gluing the head and tail of a single graph corresponds to taking the trace, and reversing a graph corresponds to the transpose (up to a change of basis).

math.CO↗

arXiv2025

The chromatic symmetric function of graphs glued at a single vertex

We describe how the chromatic symmetric function of two graphs glued at a single vertex can be expressed as a matrix multiplication using certain information of the two individual graphs. We then prove new $e$-positivity results by using a connection between forest triples, defined by the first author, and Hikita's probabilities associated to standard Young tableaux. Specifically, we prove that gluing a sequence of unit interval graphs and cycles results in an $e$-positive graph. We also prove $e$-positivity for a graph obtained by gluing the first and last vertices of such a sequence. This generalizes $e$-positivity of cycle-chord graphs and supports Ellzey's conjectured $e$-positivity for proper circular arc digraphs.

math.CO↗

arXiv2024

Adjacent cycle-chains are $e$-positive

We describe a way to decompose the chromatic symmetric function as a positive sum of smaller pieces. We show that these pieces are $e$-positive for cycles. Then we prove that attaching a cycle to a graph preserves the $e$-positivity of these pieces. From this, we prove an $e$-positive formula for graphs of cycles connected at adjacent vertices. We extend these results to graphs formed by connecting a sequence of cycles and cliques.

math.CO↗