[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-227608-en":3,"doc-seo-227608-105":30,"detail-sidebar-cat-0-en-105":97},{"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":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},227608,13056703019662,"Evangeline","https://ap-avatar.wpscdn.com/avatar/be000253a8e92610077?_k=1778726343310543188",8,"Research & Report","The minimal degree Kirchho index of bicyclic graphs - Research article","This research determines extremal results for the degree Kirchho index over bicyclic graphs. The study characterizes bicyclic graphs of order n ≥ 5 that achieve the minimal degree Kirchho index, focusing on graphs with exactly two cycles. It further derives the minimal degree Kirchho index for bicyclic graphs of order n ≥ 4 with exactly three cycles, and enumerates all bicyclic graphs of order n ≥ 4 that attain this minimum value. The analysis is based on resistance distances between vertex pairs weighted by vertex degrees.","[http:](http://www.aimspress.com/journal/Math)[//](http://www.aimspress.com/journal/Math)[www.aimspress.com](http://www.aimspress.com/journal/Math)[/](http://www.aimspress.com/journal/Math)[journal](http://www.aimspress.com/journal/Math)[/](http://www.aimspress.com/journal/Math)[Math](http://www.aimspress.com/journal/Math)  \nAIMS Mathematics, 9(7): 19822–19842 .  \nDOI: 10.3934/math.2024968  \nReceived: 22 April 2024  \nRevised: 22 May 2024  \nAccepted: 31 May 2024  \nPublished: 18 June 2024  \nResearch article  \nThe minimal degree Kirchho􀀋 index of bicyclic graphs Yinzhen Mei* and Chengxiao Guo  \nSchool of Mathematics, North University of China,Taiyuan, Shanxi 030051, China  \n* [Correspondence:](Correspondence: Email: myzmath@nuc.edu.cn)[ Email: myzmath@nuc.edu.cn](Correspondence: Email: myzmath@nuc.edu.cn).  \nAbstract: The degree Kirchho􀀋 index of graph G is defined as K f 􀀃 (G) = P d(u)d(v)rG (u; v),  \nu;v􀀒V(G)  \nwhere d(u) is the degree of vertex u and rG (u; v) is the resistance distance between the vertices u and v. In this paper, we characterize bicyclic graphs with exactly two cycles having the minimum degree Kirchho􀀋 index of order n 􀀕 5. Moreover, we obtain the minimum degree Kirchho􀀋 index on bicyclic graphs of order n 􀀕 4 with exactly three cycles, and all bicyclic graphs of order n 􀀕 4 where the minimum degree Kirchho􀀋 index has been obtained.  \nKeywords: bicyclic graph; degree Kirchho􀀋 index; resistance distance; 􀀂-graph  \nMathematics Subject Classification: 05C09  \n1. Introduction  \nLet G be a simple connected graph of order n with vertex set V(G) and edge set E (G) . Let d(vi) be the degree of vertex vi , for i = 1; 2; : : : ; n. The distance d(u; v) between the vertices u and v of the graph G is defined as the length of a shortest path between u and v. The resistance distance between the vertices u and v in G is denoted by rG (u; v) .  \nIn 1993, Klein and Randic´ [1] proposed a new distance function, resistance distance, based on electrical circuit theory. Similar to the long recognized shortest-path distance, the resistance distance is also intrinsic to the graph, and with some nice purely mathematical properties [2] . The e􀀋ective resistance is mainly used in electronic networks for which nodes correspond to vertices of G and each edge of G is replaced by a resistor of unit resistance. The resistance distance is very sensitive to small changes in the conductances, and it is suitable to discriminate between networks with similar structure [3, 4] . The resistance distance has been studied in mathematical, physical, and chemical papers [5–7], and it has important applications in chemistry.  \nA bicyclic graph is a connected graph which satisfies jV(G) j + 1 = jE(G) j. Let B(n) be the set of connected bicyclic graphs of order n. It is well known that any graph in B(n) contains either two cycles or three cycles. There are two basic types: 1-type graphs and 􀀂-type graphs; for the basic structure  \nof 1-type bicyclic graphs and 􀀂-type bicyclic graphs, see Figure 1. An 1-type graph is obtained by attaching trees Ti to some vertices of B 1 (p; q) or B2 (p; q), where B 1 (p; q) can be constructed by two vertex-disjoint cycles Cp and Cq by identifying a vertex u, and B2 (p; q) can be constructed by two vertex-disjoint cycles Cp and Cq connected by a new path v 1v2 􀀁 􀀁 􀀁 vt with length t 􀀀 1. Further, the graph S pn;q is obtained from the graph B 1 (p; q) by attaching t pendant vertices at vertex u, where t = n 􀀀 p 􀀀 q + 1.  \nFigure 1. The basic structures of 1-type bicyclic Graphs.  \nA 􀀂-graph is the union of three internally disjoint paths with two common end vertices (see Figure 2) . A 􀀂-type bicyclic graph, denoted by 􀀂pn;q;m , is a union of three internally disjoint paths Pp : v0v 1 : : : vp , Pq : u0 (= vp )u 1 u2 : : : uq (= v0 ), Pm+1 : v0w 1 : : : wmu0 , of length p; q; m + 1, respectively, with common end vertices, and the trees Tvi(0 􀀔 i 􀀔 p 􀀀 1; p 􀀕 2), Tu j(0 􀀔 j 􀀔 q 􀀀 1; q 􀀕 2), Tw k(0 􀀔 k 􀀔 m) are rooted at vi ,uj ,","cbCaiiAGtzcQC5fC","https://ap.wps.com/l/cbCaiiAGtzcQC5fC","pdf",297242,1,21,"English","en",105,"# Introduction\n## Resistance distance and Kirchho index background\n## Bicyclic graph structures (1-type and β-type)\n## Definitions of Kirchho and degree Kirchho indices\n## Prior research on extreme degree Kirchho indices","[{\"question\":\"How is the degree Kirchho index of a graph defined in the paper?\",\"answer\":\"For a graph G, the degree Kirchho index is defined as K_f′(G)=∑_{u,v∈V(G)} d(u)d(v)r_G(u,v), where d(u) is the degree of vertex u and r_G(u,v) is the resistance distance between u and v.\"},{\"question\":\"What graph class is studied as the main object of optimization?\",\"answer\":\"The paper studies bicyclic graphs, i.e., connected graphs satisfying |V(G)|+1=|E(G)|, which contain either two cycles or three cycles.\"},{\"question\":\"What are the main minimal-degree results obtained?\",\"answer\":\"It characterizes bicyclic graphs of order n≥5 with exactly two cycles that have the minimum degree Kirchho index. It also determines the minimum degree Kirchho index for bicyclic graphs of order n≥4 with exactly three cycles and provides all bicyclic graphs of order n≥4 that achieve this minimum value.\"}]","The minimal degree Kirchho index of bicyclic graphs - Research article | PDF",1789010841,53,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":92,"head_meta":94,"extra_data":96,"updated_unix":28},"the-minimal-degree-kirchho-index-of-bicyclic-graphs-research-article","",{"@graph":36,"@context":91},[37,54,74],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"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/the-minimal-degree-kirchho-index-of-bicyclic-graphs-research-article/227608/",4,{"url":52,"name":13,"@type":55,"image":56,"author":61,"headline":13,"publisher":63,"fileFormat":66,"inLanguage":23,"description":14,"dateModified":67,"datePublished":68,"encodingFormat":66,"isAccessibleForFree":69,"interactionStatistic":70},"DigitalDocument",{"url":57,"@type":58,"width":59,"height":60},"https://docshare.wps.com/thumbnails/the-minimal-degree-kirchho-index-of-bicyclic-graphs-research-article/227608.png","ImageObject",300,407,{"name":9,"@type":62},"Person",{"url":41,"name":64,"@type":65},"DocShare","Organization","application/pdf","2026-09-11","2026-09-10",true,{"@type":71,"interactionType":72,"userInteractionCount":20},"InteractionCounter",{"@type":73},"ViewAction",{"@type":75,"mainEntity":76},"FAQPage",[77,83,87],{"name":78,"@type":79,"acceptedAnswer":80},"How is the degree Kirchho index of a graph defined in the paper?","Question",{"text":81,"@type":82},"For a graph G, the degree Kirchho index is defined as K_f′(G)=∑_{u,v∈V(G)} d(u)d(v)r_G(u,v), where d(u) is the degree of vertex u and r_G(u,v) is the resistance distance between u and v.","Answer",{"name":84,"@type":79,"acceptedAnswer":85},"What graph class is studied as the main object of optimization?",{"text":86,"@type":82},"The paper studies bicyclic graphs, i.e., connected graphs satisfying |V(G)|+1=|E(G)|, which contain either two cycles or three cycles.",{"name":88,"@type":79,"acceptedAnswer":89},"What are the main minimal-degree results obtained?",{"text":90,"@type":82},"It characterizes bicyclic graphs of order n≥5 with exactly two cycles that have the minimum degree Kirchho index. 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