The issue is that we have two vertex-transitive graphs on 156 vertices, coming
from two constructions of a generalized quadrangle over GF(5).
According to sage these graphs are not isomorphic, but the graphs
we obtain by deleting one vertex from each are isomorphic. One of
these claims has to be false (in fact we know that the graphs are isomorphic).
We can supply more details about the graph(s) on request.
Krystal Guo, Chris Godsil
----------------------------------------
wombat:sgwork chris$ sage
----------------------------------------------------------------------
| Sage Version 4.3.5, Release Date: 2010-03-28 |
| Type notebook() for the GUI, and license() for information. |
----------------------------------------------------------------------
After reading in the graphs (see the file below):
sage: DualW5
Graph on 156 vertices
sage: TO5
Graph on 156 vertices
Both graphs are vertex transitive:
sage: len(DualW5.automorphism_group(return_group=False, orbits=True))
1
sage: len(TO5.automorphism_group(return_group=False, orbits=True))
1
Apparently they are not isomorphic:
sage: DualW5.is_isomorphic(TO5)
False
However their automorphism groups have the same order:
sage: TO5.automorphism_group(return_group=False, order=True)
9360000
sage: DualW5.automorphism_group(return_group=False, order=True)
9360000
(The automorphism group is the symplectic group Sp(4, GF(5)), and this is the
correct order.)
Now construct the graphs we get by deleting one vertex:
sage: DualW50 = DualW5.copy(); TO50 = TO5.copy()
sage: DualW50.delete_vertex(0); TO50.delete_vertex(0)
sage: DualW50; TO50
Graph on 155 vertices
Graph on 155 vertices
Are they isomorphic?
sage: DualW50.is_isomorphic(TO50)
True
#####
## g1 is the graph6 string for point graph of the dual of W(5)
g1 =
'~?a[~~{acbcwv_~__ooccw_faa{cf{ccaaac__bcccwoov___~____ooooccccw___faaaa{cccf{ccccaaaaac____bcccccwoooov_____...@acg_@ac...@acg?{?`a?gv_go@a...@?_ogcc?_oi@?...@?_om?_ogd?f_a@o...@{a@?_o...@?gca?@_...@o?b_@ogca?...@?gc?@{a?go`???...@ac??e?o`?cg??[?o`a?g??{?go`a???|a?_goc...@_ocgaceags?ha?_sa`ao@g?coc_n...@aop?gno@_gacoe?g?`ogaccogagoc?ha?`gorcg...@b?k@[`a?...@a@by...@?scaba?h?sa?a@gc`ch?q?c_c?c...@g_aocoa@ax?qc?_goo_...@a?oc@ca...@cga_o`?gsgpagoc_@oo_achag...@cb?_`ax?gqc?_caocg^ogac@_d?igo`?d?o_i?haoo`agoha?cc?oao`aw_q?hcacacgo`[_ocha?_ccacg...@cago`a?gcogc@?i?oqoc?ig...@cagcce?a@dbg...@c_cp?og_va_cog@d?_oa_d...@co?ah?ga@?a?_...@a@@o...@?@p...@gco_aaco_a_?`k_gcq@?caog_og...@?k?ccgh?i?aby@C?G_S@@g...@c`?oi?_d?op@g?i...@o_agcp?ag?gcpax?ca?ogsgcgcagcj`?oagcchaa?a_cg^o@cag_gcscagccg...@oa_`?`?g_oacg_`cagoao_h?a_?`axa?ogcaaop?a@?cgva...@_agg`oa_?o`|?ga?cokaa...@o`a?a?s@?hcg`?_?go`aggac?pcaog...@go`k_cq@?go_`ogcaacg...@_coocq?g?i?o`byc?g_p?o`a?h@g?_p?`oagc?gd...@_gcagdg_oa@ccpc?aoq??...@_agco____occ_@oabao...@c_h?aooc@?a`y?pc...@ooh??cog_ooag@_coap?wa?_k...@c_cq@?haach...@p?_awgac?p?ga_c_gad?i?awa?s@?K?`C_GAOGCS?@|?co...@caaoq?agcagoacog@_g...@_g@CA@@aha?ogc?w...@g?p?_?ch_`cagogacc@@ga?s?cg...@oa_cicaooc?po?og^og_@cac_cc?aop?_o...@?oagco_go_gb@?_og`ah?ca?oh?`p??cc_o?s...@o_agcai?q?_ggs?d?o`[oi?_d@@cca?cca_?_o`by?_pc?igag...@a@?ao`...@?k?__`_e?_gca@cgo`c_...@a?gaoq?@c?dca...@gco_agpa@@gaa?a_co`aw_...@?scb@?oc_...@acngoc@a?acgoca...@ca?h@g...@aoga?oog_o?g_oa?odc_ao@go...@p?_a@d??cc``o?cgao...@oa_cagca?_cwka?`?owgg?_po?i?s?h@?^o...@_aa@cag...@?_q?@h?_ocha?oqa_p?_g_o?wa?_ig_q...@a@adca?ai?ac_?qawoha?cagg_i?s?g...@ga?`[d?o_ia?`ggcs?oa_?c@?q?^...@_g_ca@c...@?ogcoh@g...@a@?gqc...@gp?_og?iggb?ocgag?co@a__qgc?...@ch@g?grag...@ggow@o?o_oga?_c...@cga_ooac?a_?a?a_ccc@?cnga...@gge@?_o...@?_?qa`a@?qc?_kggo_og...@?a?_kcgpc@g_aoagpg...@?a_a?@ggo`kh?q?c_qgaa_?gog_oa?_gg`awh?sa?`?ca...@d?i?@?qa?`b...@?o`ggaca@c...@cc_?wo`?`a?och?`oca@cog?i...@acgpcg_ao@_aaa?aa?g...@g?co`awoc?ha?ocg_?g@ogcaa...@acj_`ogacagcs?cagi?a`@?OCACG^'
# g2 is is graph6 string for point graph of T2(O) in GF(5)
g2 =
'~?a[??osr?warsetcj_qwasehoxqg`qwchk?qsefq_stiawiv?xir?kac?b?`?cocg?o...@_?`??b?`?o@_o_wco...@_?`w?e?ad_o?wcceo?wcsegagaiaa@_?aw?ok?er...@_hqxa?b@q_p...@qg_`?o?h[?gok@i...@a?beqcocg?k\ib?gocwitc?gokwiv??cgekdh_h_?cb?`?dc??_wcg?so?ap?o_?g_?d_?`?c__?d_?`?cco??@_o_xdc???wcgegg_??a?`g_aa??e?...@cc??k?cj?@@k?o?wcce?aa?g?kaib?gg_a?a?...@dc?ok?cv??eoo@?o?xk??gh`a?b?qco?gg`...@q_o?cso`?p?hso?@dcgok?iv???k...@_hqwc??_ccg?kxirg?@d?go?wys...@d?gocwitcc??a_cgekdj_@?...@a@bHQWAW?@@k??_wcg?g_?cs...@_o_@p??gg_?ca...@p??gg_?d_?`?c__?eoo?ao?o_aaw?@@k???wcgepp??gg_??b?`?pdc??aa??agacaa...@dc??k?cj?bgg?@cC??K?CJ?@@k??_e?g?kaar?pp??gg_a?b?a_oa...@dc?ococtc?gg_?cso@?o?p[...@??k__a@_?qw??or??gh`a?b?qco?gg_?cs...@_how?aig?@dcgocoitc??pp??g...@_@q...@cc??qacge?dh_o?aaw?@@ggo?wqseo?a...@d?go?wyseg?@dc??a_cgaktiaa??pp?...@a@b...@qw?o?bgg?@c?gokxir?kac?h_?cc...@_o_?wcg@p??gg_?cb?`?c...@p??gg_?ca?`?e?ac?g_?cso?ao?o_@_...@gg?@cc??k?cg??wcgor??gh_??b?...@_o`dc??aa???kacb?a?`aig?@dc??co...@_?`aig?@dc??k?cj??o?o`cc??qa??e?ad_o?wc?or??gh_a?b?_cq?b?_a...@dc?o?wcsegaga?gg_?cso@?p?psook?c?pp??g...@_?aw?ok?c?x@??k...@_?qwcg?k??gh_?cco`?@_hqxa?b??a...@dcgo?wisegoco??pp??gg`a?a@qg_`?...@dc??aacge?dj_@a...@_??bgg?@cc...@ir?@A?BO?AAW?@@ggo?wqse...@g?@dc??a_cg?k\ib?gocq??pp?...@a?bdqg_@a...@_o?aig?@d?gokwiv??...@??k__?eo?`?pchk?_sa_oi@ogd?gca...@oi?c@h?q?c_h?qoc_hgaocc?qoc_hga...@gaq?c@h?qd?gca...@oi@?gd?_sa_oka...@oi@?gd?_sa...@ohi?c_h?qoc_hgaocc@gaq?ec_h?qoc_hga...@gaq?c@xd?_sa...@ogd?gca_sa@pkg...@acgsgo`?`acicgo_?acgocgo`?o`ac`achacgo???^?????}bw????fo^???????fo?}?????bw?^?bw?????go`ao`ac`acgacgocgo`??acgo_o`adacgogo`a@acgoa...@{?n_@{?????fo?}????ofo????n_}?...@{????bw?oacgogo`a@acgsgo`?`acg?oacgo_go`ao`ac`acgac...@g???fo^?????}bw????fo??ac@{?????fo?}?fo?????^??aogo`ao...@acgqcgo_go`a?haacgogo`a@acgogo`a`acg?gq??^?bw????...@{?????fo?qc??fo^?????}????@{fo???chacgo_o`acacgog...@acgo@aocgo`?o`ac`acgacgocg...@gqfo????n_????^@{????bw??`grw????...@{?????fo?}???hao_oi@ogd?gca...@oi@?gdk_...@gaq?c@H?Q?c_H?QOC_HEW????????????????????????~~~~~'
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