English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

The complexity of graph homomorphisms has been a subject of intense study [1]&#x2013;[8]. The partition function <tex>$Z_{\mathrm{A}}(\cdot)$</tex> of graph homomorphism is defined by a symmetric matrix A over <tex>$\mathbb{C}$</tex>. We prove that the complexity dichotomy of [7] extends to bounded degree graphs. More precisely, we prove that either <tex>$G\rightarrow Z_{\mathrm{A}}(G)$</tex> is computable in polynomial-time for every <tex>$G$</tex>, or for some <tex>$\Delta &#x003E; 0$</tex> it is #P-hard over (simple) graphs <tex>$G$</tex> with maximum degree <tex>$\Delta (G)\leq\Delta$</tex>. The tractability criterion on A for this dichotomy is explicit, and can be decided in polynomial-time in the size of A. We also show that the dichotomy is effective in that either a P-time algorithm for, or a reduction from #SAT to, <tex>$Z_{\mathrm{A}}(\cdot)$</tex> can be constructed from A, in the respective cases.

Dichotomy for Graph Homomorphisms with Complex Values on Bounded Degree Graphs