{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/on-the-graovac-ghorbani-index-for-bicyclic","title":"On the Graovac-Ghorbani index for bicyclic graphs with no pendant vertices","arxiv_id":"2005.02141","date":"2020-04-30","proceeding":null,"authors":["Diego Pacheco","Leonardo de Lima","Carla Silva Oliveira"],"abstract":"Let $G=(V,E)$ be a simple undirected and connected graph on $n$ vertices. The Graovac--Ghorbani index of a graph $G$ is defined as $$ABC_{GG}(G)= \\sum_{uv \\in E(G)} \\sqrt{\\frac{n_{u}+n_{v}-2} {n_{u} n_{v}}},$$ where $n_u$ is the number of vertices closer to vertex $u$ than vertex $v$ of the edge $uv \\in E(G)$ and $n_{v}$ is defined analogously. It is well-known that all bicyclic graphs with no pendant vertices are composed by three families of graphs, which we denote by $\\mathcal{B}_{n} = B_1(n) \\cup B_2(n) \\cup B_3(n).$ In this paper, we give an lower bound to the $ABC_{GG}$ index for all graphs in $B_1(n)$ and prove it is sharp by presenting its extremal graphs. Additionally, we conjecture a sharp lower bound to the $ABC_{GG}$ index for all graphs in $\\mathcal{B}_{n}.$","url_abs":"https://arxiv.org/abs/2005.02141v1","url_pdf":"https://arxiv.org/pdf/2005.02141v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"on-the-graovac-ghorbani-index-for-bicyclic","repo_url":"https://github.com/20445/ProjetoTeste","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":0,"framework":"none","reach":null}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":null,"mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}