[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81774-en":3,"doc-seo-81774-105":30,"detail-sidebar-cat-0-en-105":92},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},81774,549758252649,"Ivy","https://ap-avatar.wpscdn.com/avatar/8000253669c5317157?_k=1778319167496531819",8,"Research & Report","A Geometric View of Combinatorial Fiedler Theory","Recently, Andrade and Dahl introduced combinatorial Fiedler theory via a parameter b(G) defined as the ℓ1-analogue of Rayleigh-quotient minimization for the algebraic connectivity of a graph. This work studies the corresponding maximization analogue, yielding an ℓ1-version of the largest Laplacian eigenvalue. The new parameter B(G) admits a precise combinatorial characterization: it equals the average of the two largest vertex degrees. A unified combinatorial and geometric treatment is developed using a cuboctahedron shell feasible set, and complexity results show counting optimal vectors for b(G) and B(G) is #P-complete.","A Geometric View of Combinatorial Fiedler Theory  \nJos´e Fern´andez Goycoolea∗ Andrea de las Heras-Parrilla† Luis H. Herrera‡  \nClemens Huemer§ Carlos Seara¶  \narXiv :2607 .005 19v 1 [ cs .CG] 1 Jul 2026  \nJuly 2, 2026  \nAbstract  \nRecently, Andrade and Dahl introduced combinatorial Fiedler theory by studying a parameter b (G) defined as the ℓ 1-analog of the Rayleigh quotient minimization characterization of the algebraic connectivity of a graph G = (V, E) . In this work, we study the corresponding maximization problem, which plays the role of the ℓ 1-analog of the largest Laplacian eigenvalue. We show that the new parameter B (G) associated with this maximization problem admits a simple exact description: it is the average of the two largest vertex degrees of G.  \nA unified combinatorial treatment of the minimization and maximization problems is presented first. Later, both optimization problems are reinterpreted in a geometrical setting. The feasible set is identified with a (n−2)-dimensional cuboctahedron shell where n = |V | . Additional structure is presented for this polyhedron, including the fact that maximizing solutions arise at its vertices and minimizing solutions arise at the centers of its facets.  \nFinally, we analyze the number of optimal vectors for b (G) and B (G) for several graph families. Although the value of B(G) is determined by the two largest degrees, we prove that counting the vectors that attain this value is actually \\#P-complete.  \nKeywords: Combinatorial Fiedler theory, Optimization, Polyhedral geometry.  \n∗ Corresponding author. Departamento de Matem´atica y F´ısica, Universidad de Magallanes, Avenida Bulnes 01855, Punta  \nArenas, Chile, [jose.fernandezg@umag.cl](jose.fernandezg@umag.cl), [https://orcid.org/0000-0001-5349-4348](https://orcid.org/0000-0001-5349-4348) .  \n†Universidad Francisco de Vitoria, Spain, and Departament de Matem`atiques, Universitat Polit`ecnica de Catalunya, Spain. Supported by project PID2023-150725NB-I00 funded [by MICIU/AEI/10.13039/501100011033.](by MICIU/AEI/10.13039/501100011033. andrea.delasheras@ufv.es)[ andrea.delasheras@ufv.es](by MICIU/AEI/10.13039/501100011033. andrea.delasheras@ufv.es)[ ](by MICIU/AEI/10.13039/501100011033. andrea.delasheras@ufv.es)[https://orcid.org/0000-0002-7219-9771](https://orcid.org/0000-0002-7219-9771) .  \n‡Departamento de Inform´atica y Computaci´on, Universidad Tecnol´ogica Metropolitana, Jos´e Pedro Alessandri 1242, ˜Nu˜noa, Santiago de Chile 7800002, Regi´on Metropolitana, Chile, [luis.herrerab@utem.cl](luis.herrerab@utem.cl), [https://orcid.org/0000-0001-7338-7611](https://orcid.org/0000-0001-7338-7611) .  \n§ Departament de Matem`atiques, Universitat Polit`ecnica de Catalunya, Spain. Supported by project PID2023-150725NBI00 funded [by MICIU/AEI/10.13039/501100011033.](by MICIU/AEI/10.13039/501100011033. clemens.huemer@upc.edu)[ clemens.huemer@upc.edu](by MICIU/AEI/10.13039/501100011033. clemens.huemer@upc.edu), [https://orcid.org/0000-0001-7557-0823](https://orcid.org/0000-0001-7557-0823) .  \n¶ Departament de Matem`atiques, Universitat Polit`ecnica de Catalunya, Spain. Supported by project PID2023-150725NBI00 funded [by MICIU/AEI/10.13039/501100011033.](by MICIU/AEI/10.13039/501100011033. carlos.seara@upc.edu)[ carlos.seara@upc.edu](by MICIU/AEI/10.13039/501100011033. carlos.seara@upc.edu), [https://orcid.org/0000-0002-0095-1725](https://orcid.org/0000-0002-0095-1725) .  \n1 Introduction  \nSpectral graph theory studies graphs through the eigenvalues and eigenvectors of matrices naturally associated with them, such as the adjacency and Laplacian matrices. An early central example of this approach is Fiedler’s algebraic connectivity a(G) [8], the second-smallest Laplacian eigenvalue, which admits a Rayleigh quotient characterization as a minimization problem; more precisely, let G = (V, E) be a simple graph, then  \na (G) = xinRV (u(xu − xv)2 : v xv = 0 ∧ v x2v = 1 ) ;  \nsee [4] for a systematic treatment. The corresponding eigenvect","cbCaicJpGzUg3DGt","https://ap.wps.com/l/cbCaicJpGzUg3DGt","pdf",806104,6,1,23,"English","en",105,"# Introduction\n## Spectral graph theory and Fiedler’s algebraic connectivity\n## Combinatorial Fiedler theory and ℓ1-analog parameters\n## Maximization counterpart and main characterization","[{\"question\":\"What parameter b(G) represents in combinatorial Fiedler theory?\",\"answer\":\"b(G) is defined as an ℓ1-analogue of the Rayleigh-quotient characterization used for algebraic connectivity, formulated as an optimization problem over a shared feasible region.\"},{\"question\":\"How is the maximization parameter B(G defined and what is its exact value?\",\"answer\":\"B(G) is the maximization analogue over the same feasible region, using the ℓ1 objective |x_u−x_v|. Its value equals the average of the two largest vertex degrees of the graph.\"},{\"question\":\"What geometric object describes the feasible set for both optimization problems?\",\"answer\":\"The feasible set is identified with an (n−2)-dimensional cuboctahedron shell, where n equals the number of vertices |V|.\"}]",1784176071,58,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":28},"a-geometric-view-of-combinatorial-fiedler-theory","",{"@graph":36,"@context":86},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/a-geometric-view-of-combinatorial-fiedler-theory/81774/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"What parameter b(G) represents in combinatorial Fiedler theory?","Question",{"text":76,"@type":77},"b(G) is defined as an ℓ1-analogue of the Rayleigh-quotient characterization used for algebraic connectivity, formulated as an optimization problem over a shared feasible region.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"How is the maximization parameter B(G defined and what is its exact value?",{"text":81,"@type":77},"B(G) is the maximization analogue over the same feasible region, using the ℓ1 objective |x_u−x_v|. Its value equals the average of the two largest vertex degrees of the graph.",{"name":83,"@type":74,"acceptedAnswer":84},"What geometric object describes the feasible set for both optimization problems?",{"text":85,"@type":77},"The feasible set is identified with an (n−2)-dimensional cuboctahedron shell, where n equals the number of vertices |V|.","https://schema.org",{"og:url":52,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":46,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":107,"slug":138},19,"General","general"]