On applications of the clique-adjacency polynomial to arbitrary finite graphs

Mathematics > Combinatorics [Submitted on 18 Jun 2026] Title:On applications of the clique-adjacency polynomial to arbitrary finite graphs View PDF HTML (experimental)Abstract:The clique adjacency polynomial (CAP), introduced by Soicher (2015), provides a powerful method for bounding the clique numbers of edge-regular graphs. In this paper, we extend the CAP framework to arbitrary finite graphs by expressing the relevant parameters in terms of average vertex degree and average edge-degree over potential cliques. This leads to a generalised CAP bound and an associated clique existence polynomial (CEP), which removes the dependence on an auxiliary integer variable and facilitates computation. We compare the resulting bounds with classical spectral and linear programming bounds, including those of Delsarte, Hoffman, and Haemers. We show that the generalised CAP improves upon these bounds for several families of graphs. In particular, we identify infinite families of edge-regular graphs arising from projective geometry for which the CAP outperforms the Delsarte bound, as well as families of regular and non-regular graphs where the generalised CAP improves upon the Hoffman and Haemers bounds. We also develop techniques for bounding feasible parameter regions, enabling practical application of the method to both structured and unstructured graphs. References & Citations Loading... Bibliographic and Citation Tools Bibliographic Explorer (What is the Explorer?) Connected Papers (What is Connected Papers?) Litmaps (What is Litmaps?) scite Smart Citations (What are Smart Citations?) Code, Data and Media Associated with this Article alphaXiv (What is alphaXiv?) CatalyzeX Code Finder for Papers (What is CatalyzeX?) DagsHub (What is DagsHub?) Gotit.pub (What is GotitPub?) Hugging Face (What is Huggingface?) ScienceCast (What is ScienceCast?) Demos Recommenders and Search Tools Influence Flower (What are Influence Flowers?) CORE Recommender (What is CORE?) arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs.