# Half-arc-transitive graphs of arbitrary even valency greater than 2 (Seminar DM)

Različica od 09:40, 22 december 2014; poglej trenutno različico
←Starejša različica | Novejša različica→

Half-arc-transitive graphs of arbitrary even valency greater than 2

Arjana Žitnik

Torek, 23. decembra 2014, od 10h do 12h, Plemljev seminar, Jadranska 19

Povzetek: A half-arc-transitive graph is a regular graph that is both vertex- and edge-transitive, but is not arc-transitive. If such a graph has finite valency, then its valency is even, and greater than 2. In 1970, Bouwer proved that there exists a half-arc-transitive graph of every even valency greater than 2, by giving a construction for a family of graphs now known as B(k,m,n), defined for every triple (k,m,n) of integers greater than 1 with $2^m \equiv 1 \pmod n$. In each case, B(k,m,n) is a 2k-valent vertex- and edge-transitive graph of order mnk − 1, and Bouwer showed that B(k,6,9) is half-arc-transitive for all k > 1.

I will present a proof that almost all of the graphs constructed by Bouwer are half-arc-transitive. In fact, B(k,m,n) is arc-transitive only when n = 3, or (k,n) = (2,5), or (k,m,n) = (2,3,7) or (2,6,7) or (2,6,21). In particular, B(k,m,n) is half-arc-transitive whenever m > 6 and n > 5, and hence there are infinitely many half-arc-transitive Bouwer graphs of each even valency 2k > 2.

This is joint work with Marston Conder.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 1, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.7116479277610779, "perplexity": 2904.9351246529154}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2020-16/segments/1585370506870.41/warc/CC-MAIN-20200402080824-20200402110824-00441.warc.gz"}