Chromatic symmetric functions of conjoined graphs

We introduce path-conjoined graphs defined for two rooted graphs by joining their roots with a path, and investigate the chromatic symmetric functions of its two generalizations: spider-conjoined graphs and chain-conjoined graphs. By using the composition method developed by Zhou and the third author recently, we obtain neat positive $e_I$-expansions for the chromatic symmetric functions of clique-path-cycle graphs, path-clique-path graphs, and clique-clique-path graphs. We pose the $e$-positivity conjecture for hat-chains.

💡 Research Summary

The paper introduces a new graph operation called “path‑conjoining” and uses it to study the chromatic symmetric function (CSF) of a wide family of graphs. Starting from the classical CSF X_G defined by Stanley, the authors recall that a symmetric function is e‑positive if its expansion in the elementary basis e_λ has non‑negative coefficients. They extend the elementary basis from partitions to compositions, writing e_I for a composition I, and show that a positive e_I‑expansion guarantees e‑positivity.

The authors first collect several known tools: the triple‑deletion identities of Orellana and Scott, the arithmetic‑progression property of Aliniaeifard‑et‑al., and explicit CSF formulas for lollipop, tadpole, and 3‑spider graphs. They also recall Tom’s elegant positive e_I‑expansion for K‑chains, which serves as a benchmark for later results.

In Section 3 the notion of a rooted graph (G,u) is defined, and the path‑conjoined graph P_k(G,H) is obtained by attaching a path of length k between the roots of two rooted graphs (G,u) and (H,v). When one of the node graphs is a complete graph K_g or a cycle C_g, the CSF of the combined graph can be expressed recursively in terms of the CSFs of the “tailed” graphs G_k = P_k(G,K_1) and H_k = P_k(H,K_1). Proposition 3.1 gives a closed‑form positive e_I‑expansion for X_{P_k(K_g,H)} involving the factor (g−1)! and a sum over e_I with coefficients (1−ℓ). Proposition 3.2 provides the analogous formula for X_{P_k(C_g,H)} with a prefactor (g−1). These formulas improve on the earlier K‑chain expansion by dramatically reducing the number of compositions that appear.

Section 4 studies spider‑conjoined graphs, where a central vertex is identified with the roots of several rooted graphs, producing a “spider” structure. The authors derive CSF formulas for these graphs and, as a special case, obtain a neat expression for pineapple graphs (a clique attached to a path). The results are distinct from previous e‑positivity proofs that relied on expressing the CSF as a sum of known e‑positive functions; instead, the composition method yields a direct positive e_I‑expansion.

In Section 5 the authors turn to 2‑chain‑conjoined graphs, where two node graphs are linked by a path and each node graph is itself a clique or a cycle. Theorem 5.1 gives a positive e_I‑expansion for clique‑clique‑path (KKP) graphs, while Theorem 5.5 handles clique‑path‑clique (PKP) graphs. Both theorems are proved by iterating the recursive identities from Section 3 and by careful bookkeeping of the composition parameters. The resulting expansions are again more compact than those obtained from Tom’s K‑chain formula.

The paper concludes with a conjecture that a new family called “hat‑chains” (a natural generalisation of K‑chains, PKPs and KKPs) is e‑positive. This conjecture opens a line of inquiry for future work, suggesting that the composition method may be powerful enough to handle even broader classes of graphs.

Overall, the work demonstrates that the composition method, when combined with the extended elementary basis e_I, provides a systematic and efficient way to obtain positive e_I‑expansions for the CSFs of many graph families. The authors’ formulas not only confirm e‑positivity for the studied graphs but also illustrate a new combinatorial perspective that could be applied to yet‑unsolved cases of the Stanley–Stembridge conjecture and its extensions.