{"id":186305,"date":"2011-05-27T00:00:00","date_gmt":"2011-05-31T16:45:37","guid":{"rendered":"https:\/\/www.microsoft.com\/en-us\/research\/msr-research-item\/on-a-conjecture-of-brouwer-regarding-the-connectivity-of-strongly-regular-graphs\/"},"modified":"2016-08-22T11:32:27","modified_gmt":"2016-08-22T18:32:27","slug":"on-a-conjecture-of-brouwer-regarding-the-connectivity-of-strongly-regular-graphs","status":"publish","type":"msr-video","link":"https:\/\/www.microsoft.com\/en-us\/research\/video\/on-a-conjecture-of-brouwer-regarding-the-connectivity-of-strongly-regular-graphs\/","title":{"rendered":"On a conjecture of Brouwer regarding the connectivity of strongly regular graphs"},"content":{"rendered":"<div class=\"asset-content\">\n<p>In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components. We show that the triangular graphs <i>T(m)<\/i>, the symplectic graphs <i>Sp(2r,q)<\/i> over the field <i>mathbb F<sub>q<\/sub><\/i> (for any <i>q<\/i> prime power), and the strongly regular graphs constructed from the hyperbolic quadrics <i>O<sup>+<\/sup>(2r,2)<\/i> and from the elliptic quadrics <i>O<sup>&#8211;<\/sup>(2r,2)<\/i> over the field <i>mathbb F<sub>2<\/sub><\/i>, respectively, are counterexamples to Brouwer&#8217;s Conjecture. We prove that Brouwer&#8217;s Conjecture is true for many families of strongly regular graphs including the conference graphs, the generalized quadrangles <i>GQ(q,q)<\/i> graphs, the lattice graphs, the Latin square graphs, the strongly regular graphs with smallest eigenvalue -2 (except the triangular graphs) and the primitive strongly regular graphs with at most 30 vertices except for few cases.<\/p>\n<\/div>\n<p><!-- .asset-content --><\/p>\n","protected":false},"excerpt":{"rendered":"<p>In this paper, we study a conjecture of Andries E. Brouwer from 1996 regarding the minimum number of vertices of a strongly regular graph whose removal disconnects the graph into non-singleton components. We show that the triangular graphs T(m), the symplectic graphs Sp(2r,q) over the field mathbb Fq (for any q prime power), and the [&hellip;]<\/p>\n","protected":false},"featured_media":196169,"template":"","meta":{"msr-url-field":"","msr-podcast-episode":"","msrModifiedDate":"","msrModifiedDateEnabled":false,"ep_exclude_from_search":false,"_classifai_error":"","msr_hide_image_in_river":0,"footnotes":""},"research-area":[],"msr-video-type":[],"msr-locale":[268875],"msr-post-option":[],"msr-session-type":[],"msr-impact-theme":[],"msr-pillar":[],"msr-episode":[],"msr-research-theme":[],"class_list":["post-186305","msr-video","type-msr-video","status-publish","has-post-thumbnail","hentry","msr-locale-en_us"],"msr_download_urls":"","msr_external_url":"https:\/\/youtu.be\/UoO3_rTqZFE","msr_secondary_video_url":"","msr_video_file":"","_links":{"self":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/186305","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video"}],"about":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/types\/msr-video"}],"version-history":[{"count":0,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video\/186305\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media\/196169"}],"wp:attachment":[{"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/media?parent=186305"}],"wp:term":[{"taxonomy":"msr-research-area","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/research-area?post=186305"},{"taxonomy":"msr-video-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-video-type?post=186305"},{"taxonomy":"msr-locale","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-locale?post=186305"},{"taxonomy":"msr-post-option","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-post-option?post=186305"},{"taxonomy":"msr-session-type","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-session-type?post=186305"},{"taxonomy":"msr-impact-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-impact-theme?post=186305"},{"taxonomy":"msr-pillar","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-pillar?post=186305"},{"taxonomy":"msr-episode","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-episode?post=186305"},{"taxonomy":"msr-research-theme","embeddable":true,"href":"https:\/\/www.microsoft.com\/en-us\/research\/wp-json\/wp\/v2\/msr-research-theme?post=186305"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}