{"id":2780,"date":"2022-08-10T17:20:22","date_gmt":"2022-08-11T00:20:22","guid":{"rendered":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&#038;p=2780"},"modified":"2022-09-06T16:14:55","modified_gmt":"2022-09-06T23:14:55","slug":"antc-talk-luis-garcia-puente-colorado-college","status":"publish","type":"tribe_events","link":"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-talk-luis-garcia-puente-colorado-college\/","title":{"rendered":"Arithmetical structures (Luis Garcia Puente, Colorado College)"},"content":{"rendered":"<div class=\"\">An arithmetical structure on a finite, connected graph G without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there is a divisor of the sum of the integers at adjacent vertices, counted with multiplicity if the graph is not simple. Alternatively, \u00a0an arithmetical structure on G is a pair \u00a0of positive integer vectors (d,r) such that \u00a0Mr = 0, where M = diag(d) &#8211; A \u00a0is a square matrix whose diagonal entries are given by the vector d, and whose off-diagonal elements are given by the negative adjacency matrix of G. Arithmetical structures were first introduced by Lorenzini in 1989; matrices of the form (diag(d) &#8211; A) arise in algebraic geometry as intersection matrices of degenerating curves. \u00a0However, they also naturally appear in the context of algebraic graph theory as matrices of the form \u00a0(diag(d) &#8211; A) \u00a0generalize the Laplacian matrix of a graph.<\/div>\n<div><\/div>\n<div class=\"\">In this talk, I will give an introduction to the topic. We will discuss some combinatorial, structural and computational aspects of arithmetical structures. In particular, we will count the number of distinct arithmetical structures on certain graph families such as path, cycle, complete and bident graphs. For paths, we will show that arithmetical structures are enumerated by the Catalan numbers. For cycles, we prove that arithmetical structures are enumerated by the binomial coefficients C(2n-1,n-1). \u00a0We will also discuss results about the associated critical group of an arithmetical structure, i.e., \u00a0the cokernel of the matrix M. \u00a0\u00a0This talk will be accessible to undergraduate students with some knowledge of linear algebra and discrete mathematics.<\/div>\n","protected":false},"excerpt":{"rendered":"<p>An arithmetical structure on a finite, connected graph G without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there [&hellip;]<\/p>\n","protected":false},"author":73,"featured_media":0,"template":"","meta":{"_acf_changed":false,"_price":"","_stock":"","_tribe_ticket_header":"","_tribe_default_ticket_provider":"","_tribe_ticket_capacity":"0","_ticket_start_date":"","_ticket_end_date":"","_tribe_ticket_show_description":"","_tribe_ticket_show_not_going":false,"_tribe_ticket_use_global_stock":"","_tribe_ticket_global_stock_level":"","_global_stock_mode":"","_global_stock_cap":"","_tribe_rsvp_for_event":"","_tribe_ticket_going_count":"","_tribe_ticket_not_going_count":"","_tribe_tickets_list":"[]","_tribe_ticket_has_attendee_info_fields":false,"_tribe_events_status":"","_tribe_events_status_reason":"","_tribe_events_is_hybrid":"","_tribe_events_is_virtual":"","_tribe_events_virtual_video_source":"","_tribe_events_virtual_embed_video":"","_tribe_events_virtual_linked_button_text":"","_tribe_events_virtual_linked_button":"","_tribe_events_virtual_show_embed_at":"","_tribe_events_virtual_show_embed_to":[],"_tribe_events_virtual_show_on_event":"","_tribe_events_virtual_show_on_views":"","_tribe_events_virtual_url":"","footnotes":"","_tec_slr_enabled":"","_tec_slr_layout":""},"tags":[],"tribe_events_cat":[13],"class_list":["post-2780","tribe_events","type-tribe_events","status-publish","hentry","tribe_events_cat-antc-seminar","cat_antc-seminar"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.2 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Arithmetical structures (Luis Garcia Puente, Colorado College) - Claremont Center for the Mathematical Sciences<\/title>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=2780\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Arithmetical structures (Luis Garcia Puente, Colorado College) - Claremont Center for the Mathematical Sciences\" \/>\n<meta property=\"og:description\" content=\"An arithmetical structure on a finite, connected graph G without loops is given by an assignment of positive integers to the vertices such that, at each vertex, the integer there [&hellip;]\" \/>\n<meta property=\"og:url\" content=\"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&amp;p=2780\" \/>\n<meta property=\"og:site_name\" content=\"Claremont Center for the Mathematical Sciences\" \/>\n<meta property=\"article:modified_time\" content=\"2022-09-06T23:14:55+00:00\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:label1\" content=\"Est. reading time\" \/>\n\t<meta name=\"twitter:data1\" content=\"1 minute\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"WebPage\",\"@id\":\"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=2780\",\"url\":\"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=2780\",\"name\":\"Arithmetical structures (Luis Garcia Puente, Colorado College) - Claremont Center for the Mathematical Sciences\",\"isPartOf\":{\"@id\":\"https:\/\/colleges.claremont.edu\/ccms\/#website\"},\"datePublished\":\"2022-08-11T00:20:22+00:00\",\"dateModified\":\"2022-09-06T23:14:55+00:00\",\"breadcrumb\":{\"@id\":\"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=2780#breadcrumb\"},\"inLanguage\":\"en-US\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=2780\"]}]},{\"@type\":\"BreadcrumbList\",\"@id\":\"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&p=2780#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Home\",\"item\":\"https:\/\/colleges.claremont.edu\/ccms\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Events\",\"item\":\"https:\/\/colleges.claremont.edu\/ccms\/events\/\"},{\"@type\":\"ListItem\",\"position\":3,\"name\":\"ANTC talk &#8212; 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