[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86015-en":3,"doc-seo-86015-105":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},86015,1099514067415,"Rowan","https://ap-avatar.wpscdn.com/avatar/100002539d78ffe74a7?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779092875211072502",8,"Research & Report","Edge transmission irregular graphs","Edge transmission irregular (ETI) graphs extend vertex transmission irregularity by assigning each edge a transmission equal to the sum of the transmissions of its two endpoints. A connected graph is ETI when every two distinct edges have different transmissions. The work proves that almost all graphs fail to be ETI, then studies order realizability questions for chemical ETI graphs. It establishes a full characterization of subcubic trees that are simultaneously transmission irregular and ETI.","arXiv :2607 . 10739v1 [math .CO] 12 Jul 2026  \nEdge transmission irregular graphs ∗ Kexiang Xu 1,2 , Ivan Damnjanovi´c†3,4, Uroˇs Milivojevi´c3 , and Sandi Klavˇzar5,6,7  \n1 School of Mathematics, Nanjing University of Aeronautics and Astronautics,  \nNanjing, Jiangsu, 210016, PR China  \n2 MIIT Key Laboratory of Mathematical Modelling and High Performance  \nComputing of Air Vehicles, Nanjing, Jiangsu, 210016, PR China  \n3 Faculty of Electronic Engineering, University of Niˇs, Aleksandra Medvedeva 4, Niˇs, 18104, Serbia  \n4 Faculty of Mathematics, Natural Sciences and Information Technologies, University of Primorska, Glagoljaˇska 8, Koper, 6000, Slovenia  \n5 Faculty of Mathematics and Physics, University of Ljubljana,  \nJadranska ulica 19, Ljubljana, 1000, Slovenia  \n6 Institute of Mathematics, Physics and Mechanics, Jadranska ulica 19, Ljubljana, 1000, Slovenia  \n7 Faculty of Natural Sciences and Mathematics, University of Maribor,  \nSlomˇskov trg 15, Maribor, 2000, Slovenia  \nAbstract  \nThe transmission of a vertex v in a connected graph G is the sum of distances from v to all vertices in G. A transmission irregular (TI) graph is a connected graph in which any two distinct vertices have different transmissions. We extend the concept of transmission to edges by defining the transmission of an edge as the sum of the transmissions of its two endpoints. A connected graph can now be called edge transmission irregular (ETI) if any two distinct edges have different transmissions. We show that almost all graphs are not ETI and then investigate several related order realizability problems involving chemical ETI graphs. In particular, we prove that for every n ≥ 15, there exists a subcubic tree of order n that is both TI and ETI.  \nKeywords: graph distance; transmission; transmission irregular graph; edge transmission; edge transmission irregular graph; chemical graph  \nAMS Subj. Class.: 05C05, 05C12, 05C92  \n∗ Email addresses: [kexxu1221@126.com](kexxu1221@126.com) (K. Xu), [ivan.damnjanovic@elfak.ni.ac.rs](ivan.damnjanovic@elfak.ni.ac.rs) (I. Damnjanovi´c), [milivojevicu@pm.me](milivojevicu@pm.me) (U. Milivojevi´c), [sandi.klavzar@fmf.uni-lj.si](sandi.klavzar@fmf.uni-lj.si) (S. Klavˇzar).  \n†Corresponding author.  \n1 Introduction  \nIn chemical graph theory, molecules are naturally represented by chemical graphs. The graph distance function is a fundamental tool for exploring these graphs, which in turn reflect the physicochemical properties of the corresponding (organic) compounds; see [23] . As the oldest and most well-known distance-based invariant (also known as a topological index in chemical graph theory), the Wiener index [26] of a graph G is the sum of distances between all unordered pairs of vertices in G. Its edge version, the edge Wiener index, was later introduced in [19] . For a survey on graphs that are extremal with respect to distance-based topological indices, see [31], and for a selection of recent developments with a focus on applications, see [9,13,14,24] .  \nLet G = (V(G), E (G)) be a connected graph with vertex set V (G) and edge set E (G), where n (G) := |V(G)| and m (G) := |E(G)| . The degree of a vertex v ∈ V (G), denoted by dG (v), is the number of edges incident to v in G. For any two vertices u, v ∈ V (G), we denote by dG (u, v) the shortest-path distance between u and v in G. A basic building block in the exploration of metric properties of a graph is the transmission of a vertex v ∈ V (G), which is defined as the sum of distances from v to all vertices in G and denoted by Tr G (v), i.e. ,  \nTrG(v) := X dG (v, u) .  \nu∈V(G)  \nThe fundamental nature of this concept is highlighted by its presence in the literature under several alternative names, such as the status [1,22] or the total distance [10,20] of a vertex. The transmission set of G is Tr(G) := {TrG(v) : v ∈ V (G)} . If |Tr(G)| = n (G), then G is called transmission irregular (TI); see [2,29] .  \nIn this paper, we extend this paradigm by introducing t","cbCaike7J1j8VS6n","https://ap.wps.com/l/cbCaike7J1j8VS6n","pdf",374292,2,1,24,"English","en",105,"# Introduction\n## Vertex transmission and irregularity\n## Edge transmission and ETI graphs\n## Main theorem and paper structure","[{\"question\":\"How is the transmission of an edge defined in this paper?\",\"answer\":\"For an edge e = uv, the edge transmission is defined as TrG(e) = TrG(u) + TrG(v), where TrG(u) and TrG(v) are the vertex transmissions.\"},{\"question\":\"What does it mean for a connected graph to be edge transmission irregular (ETI)?\",\"answer\":\"A connected graph is ETI if any two distinct edges have different transmissions, equivalently if the edge transmission set has size equal to the number of edges.\"},{\"question\":\"What is the characterization result for subcubic trees that are both TI and ETI?\",\"answer\":\"For every n ≥ 2, there exists a subcubic tree of order n that is both TI and ETI exactly when n ∈ {9, 11, 13} or n ≥ 15.\"}]",1784207811,60,{"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},"edge-transmission-irregular-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/edge-transmission-irregular-graphs/86015/",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","2026-07-26","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 is the transmission of an edge defined in this paper?","Question",{"text":75,"@type":76},"For an edge e = uv, the edge transmission is defined as TrG(e) = TrG(u) + TrG(v), where TrG(u) and TrG(v) are the vertex transmissions.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"What does it mean for a connected graph to be edge transmission irregular (ETI)?",{"text":80,"@type":76},"A connected graph is ETI if any two distinct edges have different transmissions, equivalently if the edge transmission set has size equal to the number of edges.",{"name":82,"@type":73,"acceptedAnswer":83},"What is the characterization result for subcubic trees that are both TI and ETI?",{"text":84,"@type":76},"For every n ≥ 2, there exists a subcubic tree of order n that is both TI and ETI exactly when n ∈ {9, 11, 13} or n ≥ 15.","https://schema.org",{"og:url":51,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":51},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,109,114,119,122,127,130,134],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":29,"slug":108},5,"Comic","comic",{"id":110,"doc_module":4,"doc_module_name":46,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":46,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":46,"category_name":124,"show_sort_weight":125,"slug":126},9,"Religion & Spirituality",20,"religion-spirituality",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":128,"show_sort_weight":125,"slug":129},"World Cup","world-cup",{"id":131,"doc_module":4,"doc_module_name":46,"category_name":132,"show_sort_weight":131,"slug":133},10,"Lifestyle","lifestyle",{"id":135,"doc_module":4,"doc_module_name":46,"category_name":136,"show_sort_weight":106,"slug":137},19,"General","general"]