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arXiv · 2609.13888

Independence polynomials and the weak Lefschetz property for tadpole graphs

Tran Quang Hoa·Nguyen Duy Phuoc·Tran Nguyen Thanh Son
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Abstract

Let $T_{m,n}$ be the tadpole graph obtained by joining a cycle $C_m$ to a path $P_n$ by a bridge. We prove that the independence polynomial of every tadpole graph is unimodal and establish sharp bounds for its mode. The unimodality result follows from a general criterion for graphs obtained by attaching a path to a fixed vertex. Over a field of characteristic zero, we also give a complete classification of the pairs $(m,n)$ for which the Artinian algebra defined by the edge ideal of $T_{m,n}$ together with the squares of all variables has the weak Lefschetz property.

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Tran Quang Hoa, Nguyen Duy Phuoc, Tran Nguyen Thanh Son. 2026-09-12. Independence polynomials and the weak Lefschetz property for tadpole graphs. https://arxiv.org/abs/2609.13888

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