# SageMathCell online
# https://sagecell.sagemath.org/?q=bahvid
# 11 isomorphic graphs found 
#  Z_11*(9)Z_23  F. Comellas, M. Mitjana (August 1994) 
#  Z_11*(3)Z_23  M. Sampels (1997)
#  all others by F. Comellas (2024)
# The graphs are in sparse6 format .

GCM=Graph(r":~?B|_C?_C?_C?_C?_K@_K@_K@_GE_SA@CA_SA_SA_[B_[B_WD_[B_cC_cC__F_cC_kD_kD_kD_sE_sE_sE_{F_{F_{F`CG`CG`CGbkMA_mFW|d?|`_]FkuFkIDW|`gwFkQFk^FstFsfFpBew~GcTCGqF_~`WbF{kF{UC_~c?sF{RGCfEH?fH?GSsGCeDx?a?vGK[FP@`wUGKcCh@bgkDh@c@@dPAaGYGSsGSaCpAe@Ba@BbW`G[ZEXBbgibgcbO\FK\bkHGclE`CcWtGchEPCbGcCgpGcQFPEHX^b@QL[HC?cJPkdPDIXka_aJhk`oxKxk`HHLcNEPdL[bJHjMCmE?uLkYDgqF@lcxTK`lbGpHPm`w{L@mL{RCHcLsI@g^LsWD@eb@PLKWGxLb?XDXWb?wIP]b?`CXT`_oH@n`WdK`n`OfDHVL{kK@od@oagqI@UMCJEXQJCKIPgao[ISrFXpcgtJpcMK^EpdMKRJXaMKjHxWKHpd_qEhMNSZFXTJA@P{yJ`zP{JB_cFYNeOvGxaPKKJpaKyOdwpJH|QCOHx`MaPd`FLQKQKVDxIIqP`oSIaP`whHpYPSXF`MMXuexMHx\OCVHpfNQDeo{HpUNkHD`DKCSD`XJX~egwFKtJStH`aN@}aodDorKXyOCPChLIHya?jDxdOiQaOTCxbQSeDOsJi?QShF@aKhvRKQI@|OA^TKQDpPJAhe@aKprOydbOoGiDOyJTSLCP`RYiao[DGwMYM`gTBohM`}a?hGpJNcbGxZMy@S{UDPeMyCRs]DhvPYXSSaF@JOQ\cxdKxga@dOKTC@dOyVRcjHh^KYGRI]dWrH@uNIX`ggJiHSQjaxHH`XPAj`P[KaCRqjcG{JXfOyHQqp`_yF`_SQcSsMCopF`VaWdD@ZMPt`oZH`RJY[eprOQURy{`wXHh[PQUSKvJIHPYKdAHR{XBx\MqEPAp`H\KYTRAp`o`JQCSiva_XCGnOY`fHDLHiUyxV[IGp`LP~dXIJQAQYSc@EI@zOQrb_aKqdSs^HxSOaTSZ@awvFYCPAmUB@axEIq@SIyWsLBPJHxUSAufPgPiMWrFXKSCpqPq\cXPIyIVQzdPSNa`TqwXkQBo^Kq`WBAXkPBpqNq^UrG`_MDPtNqtWkJAHhPagYBQ`PJQYgUcuNH~SzGXKRAooJ`{Ur?e@HIXSPQuXbMaglGhEJaSRbNb_mHALVjVZKOA_bMikVkRHPLKABWjK`wcK@zNafWjTfPKJpqOYlUZU`GJExPJzIZsgEWuHYSXSxHPsMq_XzTaWxGxWRQ{Xz[c@hOiRSA~Wb\dozKpxOrR[[TBXgMaTUjUZcVCxTPiaTatX[QJyIRAbZRgbWlH`OPamZja`W]DgrJH|UzSaw[C@OJpwWBfcOmIX]KYZWrZbO]IHTNADQzKaGpIx^NYDPB```RNH|QyaTzTbxGKH~OqYSRee`DNYWRYnUz[`GeIa\SYqUzjeGsFPcRQoWra`OzNasVBBWrmaGnKHcLIqWRf`gYConHALWZM~~~") 
Gsamp=Graph(r":~?B|_C?_C?_C?_C?_K@_K@?c@_K@_SA_SA_SA_OD_[B@CB_[B_[B_cC_cC_cC_kD_kD_kD_sE_sE_oF_sE_{F_{F_{F`CG`CG`CG`O_eG|d?|ao[FkwFkmF_|bW|cC_E[OC?`c@?cCXBg_aOvFW}dW}aO}`gWFseE?}aW]Fp@b?aF{MDO~G[lF{TF{PE_~awYDg~eP?`goGCJDgnGCKFH?bP@dH@`waDOrGKjGK`Ep@`_XGSQA_bGSdGSIGSPCPA`OxFSIC{I`GIDcI`gRBwfG[tG[zG[TG[HC`BfPDJCXCp\LsgHxm`_SLPmbXaLsXD`SLs\JC[COcE`IJC^CWtJ@ofPJLHlfP_aGTFPJeoyH@PbOyH`UMKQIHXL{^DpneHeL{HDpOIXnahdMCRF`GMCVDpEKXo`wjGxjMC\DWoIXpb?fJppb_iL@p`wvGxQMSUCHJHhqeOvJXqaOoGhqcgpEpXMShGhjLcjGkLAXCGh`bPDchGH`VJ`rcXlM[OFHMM[MG`YM[JLaAa_wIQScX\NYSagiD`FHiSbwfIaIQc\FXQKH}QcQD@XJPeMyAaovIxfOQUdWqL`wQARe_xLaGcOpHpkNCXDwuExgLcXEpJOYTbpVKQTchcPiTcotG`aNaT`wrHqUbWnHaUa?RDXOKPtPYU`geEo{Ih_Q{cD?qJyNQ{HB`NN@yRCJF`JRCQJXcLABRCJJqPQSJ@whKCJH@KMaH`W]I[KDxZNI@RKUEqRRKODHLLP|a@MJx`Mpya@VJ`}TSOIhlNiHSS\K@dQQ\UC]KpjQQhePQKXgLINQSbIPgNAq`pUOIIUSwKxsPQ_USZF@GL@~PArehYJhdOynU[gG`zRi^UcZGpwUcMCXJOiKPysfXEOqp`wVCwnGpx`odCpEIsMGxbOi]SitbgcHxgUkkHP]NQ``_sHQ@SQ~bOfDHINyRSKrEgzIiGUzC`GzIqLQI}fXdNiJPyOT{NAp|Py\WsqHI?SiiWsVChiPqiVJEdwvI`iPybTKSBwrExfObBagdJhsUywW{UDPHNAkTzF`gMF@HJXtOCHAGiFHHMqJXC]KHvOq^VShE`FKpvSk\IX[JpvSYyb?oG`lOAlUJPb?rLi`TYnUBBaGlKicVzTd_tI@ZTq~XbTd`bLH}TyzVsSAxMJZIXZUdhNJy\WJBYslH@HLQDRQcdgpG`^LYZTZVaGWJpeMqLRcKCpTOq[VZLcG{IHsQYcWj\boqE``Kx{OjAc_mNYFUzR[KRDpcMaFQJ?[SpIIEPiZTiuf@FNygUrMZSaDPUKhePIMSCTBxYKRIZzfaw^H`aPapYRgaWXCGoKagTjHcwrJH^OALTqxehOIxxPQkWj]`G[CG{H`QPa|bPfNa@RqiWJdbW[D@CNH|TbOc_xIYMSAdVzi`ggI`\Ma]SJ@aGeE?sJPjTqxa_VC_xH@\NqyeGwF`eOaPWRKa_nHhOJPjSRLbW]CPcLQLSypdHLKXyNaCQJYd`SIpXPalVbDbO`KH}OABTbNb_bIYHSQcWJ[~~~") 
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G16=Graph(r":~?B|_C?_C?_C?_C?_K@_K@_GF_K@_SA_SA_SA?sA_[B_[B_[B_WD_cC@CC_cC_cC_kD_kD_kD_sE_sE_sE_{F_{F_{F`CG`CG`CG`gRf_|bwlFkZBo|cG{FkVB__FkhFkRCSRA_taWXaWdEoxGSRFSRCOmFCMBGYFsWC_}dW}cgqFsbFscD_}`gc`ge`gX`gw`gkGCYFxB`G~`GtF{gE_~coiEG~cx?``?eH?ex?bgmGKiGKtGKSGKjGKQGSfFOzGSnGSbG[XFHBcWpEXBcOoG[PDxB`OdIh_bwdISdDGvIsdDhOI[IBWdE[nHhVa_[HhMaxLJPgLsNApLLK`HhVK{J@_WGxXbobDXF`GkE@FJxke_zG`jb_kG`ja?YJ`dL[]Eh\KXja`ZLcpH@k`opKXk`WmKpkbO`CPSLkUB_cE`lagsHpleH]LkQH`mehZLHiLsMKHmfPWL{_J@bL{PIxaL{oHX`L{I@onGhYOKQEpTJSHBXYKpwagxJP_Mk_GhQJSfKHxcwgE@MI[YCxONaGPsfHHNPKfEXUNKoGpGIHqN[OBxWJpq`WvG`^KPqPcWIX^MAJb_^NyCP[KHp|PcmF@bMYKcHEOyKa`NIXXKiLd?{KiLdXbOALbgaHaM`w[NiHPQMfPhNQN`_lNQCP{PF@JNIBP{eF@aOQN`OVH`{dxCIP{O{KF?{L@{OITdguMH{cWyJYARCXC_gFHZJ`adPZMacc?mJXwRCqKPrMqQRKvGpeMABQQea_vFHPQCSDXRIi[SCcD@GRYecGsKyCQYeePUSQedOxHpzOqfb?hJaES{H@wyJqCS{UBhSIyg`ozLIHTKMH@_SAha?eHXfRA^TKJBpfOY]TK\FXDIAYcOuKHoRQadGkH@fPAYRimaGUAx_QIYb@WLH|Rqo`wjJh|QazcGlDx|UCKKaBQYUUKKEPeOyORy|`OxHp]Uawe@XJ`rUY|a?rIpdSAkVkTD@\LAXVsTBXPNq}boeDorHQoUz?bwrGq^Ur?fPSLPpMiqaxVOQOUQxbWrJxrSYqUjCcXPNQHRYubGbIp_KIZbPGOi^TbE`wgD_zGhgOCWKPzNqRXBLc@INAEVBAXk{J`~PApXsOAO{IPuXs_C_qIivXSPI@xRqiW{_ExQQYyWzHYSMJ@uSarXSHHH[JxiMIGaOoIIBRIkYkhNq[SZEZKhEguHPKQrX`OXCOlJHcY{NAolEpeTbSYsvG`sNqIQIbaOWBYsVRRaxHHx_OzAZj]aPeMASSijVSUFXKIHQZciDWmI@sOIa[[UEHKOyOTi{bguHxiMracH\JptQa\VJe`W\HxbSQiWZKawiIaIQrdaHUNy?QilUjhdPEIaCVJ@YJWbPOJhvSjg`HJNA[TrBXJkaGqE`\My_YB_conGasWbPYrnehJK`dNyVVBZ`W]E_zKHoYR]a?[CokOiERan`w^GpdLQAQBbag]HPRJy`UbDePHIiEPQ]TRe`OTK`sSiuWJhf`DOiSVbIYBkdHpSIlXbVZZn`WOBWwQJBZB^~~~")
G18=Graph(r":~?B|_C?_C?_C?_C?_K@_K@_K@_GD_SA_SA@CA_SA_[B_WE_[B_[B_cC_cC__F_cC_kD_kD_kD_sE_sE_sE_{F_{F_{F`CG`CG`CG`GJCCfEGrFkSDGmE_|ew|aG|a_XFkRAw}bW}`gVCWwFsjFp?`_YBW~bo^FW~bgyF_~cC_GK_c?xc?uc?cDkHA?T`KHA{HCsH`KWCP?cgtGCQE@?`OQGCUGC[D`@fX@eOxGK^DX@`wYGKJCSJAsJAgcD{JFSJBCJAOiFSMBxAehAcOwGS`EhAd@Aaw[Ex_KSVDHXJpbaws`oVDGoH[SAojHPj``LL[fHPSKpjdwpH`LLcjI@PLHk`PNL@iLcUISlIPlcGnISzIPfepQKkcChQIh\`wmGXGcHBdOqGXI``BI{KD@B`PMJ`l`wxLHlbp`Kxld@PJXl`wTGpKLkODgnGhHaGQEhDH@TcoiD`DGxYLCrIHma@cLs{IXcLsLBHIHqEbWwHPXO[OEPIK[rHPXJxucOhHHZMiJagpNAJextNyGP[PK@vNAKepZNX{PQK`ooG`yPcIHH|aHRNkzLH|PKRDxTIp|OStIXnb?dEGyL{NEHWL{kG`xOIL`_cGpfPQLcpWNa?Pk[GpGMyLaGmJhpMcRE`FK`pM[MBhVJP[MHqcoxFXaNaMcWmPsbF`iNqM`OfHXtNI`bXLKptRsmF@`MkZEOtKPtOclH@_QyebGcQiZSYe`gcCgsQqVS{jKi?RIdS{]EX^MAWS{UFXCMcUE_uNsULIISckK`}P{YFHEKANaOiJ`oRigb_pIiHSigbO^IyXRagb`MMYUTCeE`]OQPQcSCXbMQPQ[WBgrHy@QIQf`SJqhbokE@SLQYTKiExCMqTVKTD`LNAGQiuep]KqDQkgHh`KaEQi^`g]JhrUI|ehJIinUY|cxTIpbUApVseHHhMAS`O`CX[RSI@g[H`fcHFJQ@QAtV{RChMKAcVa~b?zH`PUQsV{OIxbOYjTshF@SQQjTkaE@OJ`^Nz?bGqJPoTzJcHdOYGSAn`wbJ@wOi`TyvcgfF`OOQRXkRBxcMrCWrLdhVMaIQAmaOaG`RIa@cxCHpvPK\IXqRiiXBM`gSIpxVZKXsTB?mFA]WSkJHeNykWS\HX`QBCZCYEpEIaVSAxWcOExULPzTjXa_bEOxIJQYbXagWJiSSiiexoNIUSceHx}QQtWJUbWlJaGRZBYK[H`dNypVBJYShDhSKHfNb\ZsXDQOUAzf`MMyFRZIZB\bpXJh~OiHRJbd?yHXYTS]LPxQqaUZV[CZIAARY_Saq[s\J@gMr@bwiE?xIAoVREa@LNYrVAzWjh`oQBOqHzOc_yIHYJxfNZSd?jDpwWrIYZkeXFH``NRZbgjIhgNaCPJ]``WJX]QzSYzn`oRHxZRI^SbbbO^DwpOABUJ]a_rNQ?UQvVZeaG{H`PKA[Rrbd?hEpaRA[Wjh`gsLPvNasUrecwoKYURJFXJhcGvF@^UAqYjkbGyIXeObDZzncgnHXgNqGRzR`_MAHUMqYSYv~~~")

# List of graphs to process
graphs = [
    ('GCM   ', GCM), ('Gsamp ', Gsamp), ('G2   ', G2),('G3   ', G3),('G4   ', G4),('G6   ', G6),
    ('G8   ', G8),('G12   ', G12),('G13   ', G13),('G16   ', G16),('G18   ', G18)]

def check_all_isomorphisms(graph_list):
    n = len(graph_list)
    print("\n Isomorphism check of all pairs (a dot means the pair ARE isomorphic):")
    for i in range(n):
        label_i, G_i = graph_list[i]
        for j in range(i + 1, n):
            label_j, G_j = graph_list[j]
            if G_i.is_isomorphic(G_j):
                print(".", end="")
            else:
                print(f"\n{label_i} is NOT isomorphic to {label_j}")
                
    print()  # Newline at the end

def count_k_cycles(G, k):
    count = 0
    visited = set()
    def dfs(path, start, depth):
        nonlocal count
        current = path[-1]
        # Early exit if we’re going too deep
        if depth == k:
            if start in G.neighbors(current):
                # Normalize to avoid duplicates
                cycle = tuple(sorted(path))
                if cycle not in visited:
                    visited.add(cycle)
                    count += 1
            return
        for neighbor in G.neighbors(current):
            if neighbor not in path and neighbor >= start:
                dfs(path + [neighbor], start, depth + 1)
    for v in G.vertices():
        dfs([v], v, 1)
    return count   # each cycle counted twice (once forward, once reverse)

def algebraic_connectivity(G):
    """
    Compute the algebraic connectivity (Fiedler value) of a graph G.
    INPUT:
        - G: a SageMath Graph
    OUTPUT:
        - The second-smallest eigenvalue of the Laplacian matrix of G
    """
    L = G.laplacian_matrix()
    eigenvalues = L.eigenvalues()
    eigenvalues.sort()
    
    if len(eigenvalues) < 2:
        return 0  # Trivial case: empty or isolated vertex graph
    return eigenvalues[1]


def domination_number(G):
    """
    Compute the domination number of a graph G using MILP.
    INPUT:
        - G: a SageMath Graph
    OUTPUT:
        - The domination number (integer)
    """
    p = MixedIntegerLinearProgram(maximization=False)
    x = p.new_variable(binary=True)   
    # Objective: minimize the number of chosen vertices
    p.set_objective(sum(x[v] for v in G.vertices()))
    # Constraint: each vertex is dominated
    for v in G.vertices():
        p.add_constraint(x[v] + sum(x[u] for u in G.neighbors(v)) >= 1)
    return p.solve()


# Print properties for each graph in the list
print("\n Main properties of the graphs\n")
for label, graph in graphs:
    print(f"{label} | Order: {graph.order()} / Size: {graph.size()} / 8-reg.? {graph.is_regular(k=8)} "
          f" / Diam.: {graph.diameter()} / Avg.dist: {graph.average_distance().n(digits=6)} / Girth: {graph.girth()} "
          f" / Alg.connect. {algebraic_connectivity(graph).n(digits=6)}  ") 
         #   Domin. number: {domination_number(graph)}"
        
print("\n Symmetry properties of the graphs\n")
for label, graph in graphs:
    print(f"{label} | Aut.group.ord.: {graph.automorphism_group().order()} /  Cayley ? {graph.is_cayley()} ---  vtx.trans. ? {graph.is_vertex_transitive()} -- edge.trans. ? {graph.is_edge_transitive()}" )
    
# Check isomorphisms
# print(f"Are isomorphic G5 and G6? {G5.is_isomorphic(G6)}")

check_all_isomorphisms(graphs)

# Compute the distance distribution from a given vertex v in graph G
# Returns a list where the i-th element is the number of vertices at distance i from v
def distance_distribution(G, v):
    from collections import Counter    
    distances = G.shortest_path_lengths(v)
    distribution = Counter(distances.values())
    result = [distribution[d] for d in sorted(distribution)]    
    return result

print("\n")
for label, graph in graphs:
    print(f"{label} distance distrib. from vertex 0: {distance_distribution(graph, 0)}")


# Counting k-cycles for each graph
print("\nNumber of k-cycles for k=3 up to 6")
for label, graph in  [('GCM   ', GCM), ('Gsamp ', Gsamp), ('G2   ', G2)]:
    print(f"{label} ", " ".join(str(count_k_cycles(graph, k)) for k in range(3, 7)))
    
##