[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85196-en":3,"doc-seo-85196-105":30,"detail-sidebar-cat-0-en-105":91},{"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},85196,962075114101,"Seraphina","https://ap-avatar.wpscdn.com/avatar/e000253a75eb197efd?x-image-process=image/resize,m_fixed,w_180,h_180&k=1780044092746381165",8,"Research & Report","Homological Invariants of Edge Ideals of the Multiple Extended Complete Split-Like Graphs","Homological invariants are developed for the edge ideals of multiple extended complete split-like graphs MECSab,n, constructed by attaching an independent set of size a to n disjoint copies of the block Kb+K2. For the case n=1, prior one-block results are recovered. Using Hochster’s formula, tensor products of minimal free resolutions, and Betti-number formulas for graph joins, explicit descriptions are obtained for the independence complex, independence polynomial, Hilbert series, Betti strands, regularity, and projective dimension. The study also classifies well-covered and unmixed cases, shows the family is never Cohen–Macaulay, computes induced matching numbers, and records procedures for evaluating Betti data.","arXiv :2607 . 10300v1 [math .AC] 11 Jul 2026  \nHomological invariants of edge ideals of the multiple extended complete split-like graphs  \nBilal Ahmad Rather  \nSchool of Mathematics and Statistics, Shandong University of Technology,  \nZibo 255049, China  \n[bilalahmadrr@gmail. com](bilalahmadrr@gmail. com)  \nAbstract  \nWe study the graphs MECSab,n  Ka∇􀀀n(Kb+K2 )􀀁, obtained by attaching an independent set of size a to n disjoint copies of the block Kb + K2 . For n = 1, we get MECSab, 1 , and recover the results of one-block case studied in [Anand, Gupta, Rather and Singh, Homological invariants of some complete split-like graphs, Beitr. Algebra Geom. (2025)] . Using Hochster’s formula, tensor products of minimal free resolutions over disjoint variable sets, and the Betti-number formula for graph joins, we derive explicit descriptions of the independence complex, independence polynomial and its analytic properties, Hilbert series, linear and quadratic Betti strands, regularity, projective dimension, and several structural invariants of MECSab,n. We further classify the well-covered and unmixed members, compute induced matching numbers, show that the family is never Cohen–Macaulay, and record algorithmic procedures for evaluating the Betti data.  \n2020 Mathematics Subject Classification. Primary 13F55; Secondary 05E40, 13D02, 05C75, 05C69 .  \nKeywords. Multiple extended complete split-like graph; join of graphs; edge ideal; graded Betti numbers; Castelnuovo–Mumford regularity; well-covered graph.  \n1 Introduction  \nThe passage from a finite simple graph to its edge ideal has become one of the most productive interfaces between combinatorics and commutative algebra. If G is a finite simple graph with vertex set V (G) = {x1 ,..., xr }, then the square-free quadratic monomial ideal I (G) = 􀀀xixj : xi , xj ∈ E (G)􀀁 packages the adjacency data of G into a form accessible to homological methods. The minimal graded free resolution of R/I(G), where R = K [x1 ,..., xr ], encodes graded Betti numbers, projective dimension, regularity, and several finer measures of complexity. This circle of ideas goes back to the foundational work of Hochster, Stanley, Villarreal, and many others, and remains central in current studies of monomial ideals, simplicial complexes, and graph parameters [11 , 13 , 18 , 24 , 26] . The attraction of the subject is that algebraic invariants can often be computed from graph-theoretic information, yet the resulting formulas are rarely obvious from either side alone [6 , 7 , 9 , 10 , 12 , 14–16] .  \nHomological invariants of multiple extended complete split-like graphs 2  \nAmong graph classes that are particularly suitable for this interplay, split graphs and their extensions occupy a useful middle ground. On the one hand, they are structured enough to admit exact enumeration arguments. On the other hand, they are rich enough to exhibit nontrivial homological phenomena. Complete split graphs, nearly complete split graphs, and related families have already been examined from the viewpoint of graded Betti numbers, regularity, and projective dimension by several authors [1 , 3 , 8] . Recent work also points to broader connections with domination parameters, shellability, and Cohen–Macaulay type behavior [5 , 19 , 21–23 , 27] . What emerges from these studies is a recurring pattern: once a graph family is built from a few elementary pieces by join, disjoint union, or Cartesian-type operations, the corresponding algebra tends to inherit highly organized syzygy patterns.  \nSuppose G 1 = (V(G1 ) , E (G1 )) and G2 = (V(G2 ) , E (G2 )) are simple graphs with disjoint vertex sets, that is, V (G1 ) ∩ V (G2 ) = ∅ . We use the following two standard constructions. The join of G 1 and G2 , written as G 1 ∗ G2 , is the graph whose vertex set is the disjoint union V (G1 ) ⊔ V(G2 ) , and whose edge set is obtained by taking all edges already present in G 1 and G2 , together with every edge joining a vertex of G 1 to a vertex of G2 , th","cbCaiiZ1a9JdiZvn","https://ap.wps.com/l/cbCaiiZ1a9JdiZvn","pdf",394001,2,1,28,"English","en",105,"# Introduction\n## Graphs, edge ideals, and homological invariants\n## Join and disjoint union constructions\n## The MECSab,n graph family and its motivation\n## Anticipated new homological behavior","[{\"question\":\"How are the multiple extended complete split-like graphs MECSab,n constructed?\",\"answer\":\"They are formed by taking an independent set of size a, attaching it to n disjoint copies of the block Kb+K2, and joining every independent-set vertex to every vertex in every block copy.\"},{\"question\":\"Which mathematical tools are used to derive the homological invariants?\",\"answer\":\"The work uses Hochster’s formula, tensor products of minimal free resolutions over disjoint variable sets, and the Betti-number formula for graph joins to obtain explicit invariant descriptions.\"},{\"question\":\"What structural conclusions are obtained about Cohen–Macaulayness and related graph classes?\",\"answer\":\"The family of these graphs is never Cohen–Macaulay. The paper also classifies which members are well-covered and unmixed and computes induced matching numbers.\"}]",1784201677,71,{"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":86,"head_meta":88,"extra_data":90,"updated_unix":28},"homological-invariants-of-edge-ideals-of-the-multiple-extended-complete-split-like-graphs","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,47,50],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":20},"https://docshare.wps.com/document/","Document",{"item":48,"name":12,"@type":43,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":43,"position":52},"https://docshare.wps.com/document/homological-invariants-of-edge-ideals-of-the-multiple-extended-complete-split-like-graphs/85196/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-24","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"How are the multiple extended complete split-like graphs MECSab,n constructed?","Question",{"text":75,"@type":76},"They are formed by taking an independent set of size a, attaching it to n disjoint copies of the block Kb+K2, and joining every independent-set vertex to every vertex in every block copy.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which mathematical tools are used to derive the homological invariants?",{"text":80,"@type":76},"The work uses Hochster’s formula, tensor products of minimal free resolutions over disjoint variable sets, and the Betti-number formula for graph joins to obtain explicit invariant descriptions.",{"name":82,"@type":73,"acceptedAnswer":83},"What structural conclusions are obtained about Cohen–Macaulayness and related graph classes?",{"text":84,"@type":76},"The family of these graphs is never Cohen–Macaulay. 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