# SageMathCell online    https://sagecell.sagemath.org/?q=bqmdqx
# 

'''

 Main properties of the graph
H5K4p    | Ord.: 7932 / Size: 23796 / Diam.: 6 / Avg.dist: 5.27214 / 6-reg.? True /  Girth: 3   

 Symmetry properties of the graph
H5K4p    | Aut.group.ord.: 1 /  Cayley ? False ---  vtx.trans. ? False -- edge.trans. ? False
Degree histogram H5K4p   : [0, 0, 0, 0, 0, 0, 7932]

H5K4p    distance distrib from vtx. 0: [1, 6, 30, 150, 750, 3270, 3725]

Number of k-cycles for k=3 up to 4
H5K4p     160  &  40

'''

import networkx as nx

H5K4p = Graph(r":~@z{nO_???@??_?K?CNOW???D?A_?{?SNNG???E?B?@G?WNNW???F?B_@??VNNg???G?Ao@K?UNNw???H?BO@C?TNOG?CJVDK@wcf\Keg?CJWDJPw[f`Kew?CJXDJpwSf_KfG?GIdDQ@uwgPKXw?GIeDQ`uogOKYG?GIfDPpu{gNKYW?KI}C~@z[fyK`G?KI~C~`zgfxK_G?KJ?C}pz_fwK_W?OIJDDpxwgiKSG?OILDD`x{ggKSg?OIMDE@yGgkKRg?SAW@Jo[WHWMoG?SAX@J?[OH[MoW?SAY@J_[GHZMog?WRJEhqGKhYMbg?WRKEhAGCh]Mbw?WRLEhaF{h\McG?[M^FNP}smELWg?[M_FM`}kmILWw?[M`FN@}cmHLXG?_OtD]qY{joL}G?_OuD]AYsjsL}W?_OvD]aYkjrL}g?cKFECaP_o]Krw?cKGEBqPWo\Kqw?cKHECQPco[KrG?cKIEBaP[o_KrW?cKJECAPSo^Krg?gBI@DO\sGgM{W?gBJ@Do\kGfM{g?gBK@D?\wGeM{w?kPMDjAVkjXMCG?kPNDjaVcjWMCW?kPODiqVojVMCg?oQXEnqE_iMMUw?oQYEoQEWiLMVG?oQZEnaEciKMVW?sKxD}AQ{nmL?G?sKyD|QRGnlL?W?sKzD|qR?nkK~W?sK{D}QQwnjK~g?sK|D|aRCniK~w?wMEFAa@sm`LQG?wMFFBA@km_LQW?wMGFAQ@wm^LQg?{Ap@VoX?HBMvG?{Aq@WOXKHAMuG?{Ar@V_XCH@MuW@?NPFGa?GlWLdw@?NQFHA?SlVLcw@?NRFGQ?KlULdG@CK_EOaM[oCKxg@CK`EPAMSoBKxw@CKaEOQMKoFKyG@CKbEOqMWoEKxG@CKcEOAMOoDKxW@GQqE[qJChvM]G@GQrE\QJOhuM\G@GQsE[aJGhtM\W@KOBDcaXGkeLqW@KOCDdAXSkdLpW@KODDcQXKkcLpg@OBa@Q?YcGNNBG@OBc@PoYgGLNBg@OBd@QOYsGPNAg@SLQEIANwnSLDw@SLREIaNonRLEG@SLSEHqN{nQLEW@SLTEIQNsnPLEg@SLUEIqO?nTLDg@WO[DoqUKkILwW@WO\DpQUCkHLwg@WO]DpqUOkLLvg@[MwEzqB[lmL^W@[MxE{QBSllL^g@[MyE{qB_lpL]g@_Q?EaqHkieMPW@_Q@EbQHcidMPg@_QAEbqHoihMOg@cA^@EoX[GRMig@cIOD?`vOfeKLg@cKOD~QMgnVKlg@cMKE|P~GlZLKg@cOGD_QUWj^Ljg@cQCE]QEwhbMIg@gA|@To^CFtMqG@gIPD?pvSfaKLg@gKmEMQSOmxKtG@gNGEtq@?mtLZg@gPaDfqS_kZMAG@gPeEsqIgi@Mgg@kBZ?~OZSHlMxg@kIQD@@vCfbKLg@kLKDvqO_opK{g@kLmFRqAwlxLig@kOeDnQ[?j@LrG@kR]EdqD?i^M`G@oBx@MOVcHNN@G@oIRD?@vGfcKLg@oLiEEqKooRLCG@oMiFKQCok|LSG@oNiDuqYGj|MHg@oR?EUqGoi|MXg@sA@@\O\KGpNGg@sIND?PvKfdKLg@sJqETqQWntLJg@sNeFCp|OmVLbG@sPCDWqWOkxLyg@sQaElQK_hDMQG@wAZ@F?X_GSMig@wImDN`{wfGKTG@wKjELqSKmxKsG@wMfFLACkk|LSW@wOgDmqZ{j@LrW@wQcEkqK[hDMQW@{Ay@TO^?FtMpG@{IoDN@{sfGKTW@{LIDvqOgooKzw@{NdFDP|OmTLag@{NiDuaYSjyMHW@{QDE]aE{h^MIgA?BX?~OZ[HkMwwA?IlDNp{ofGKTgA?LhEEAKooULBgA?MLE|`}wl[LKgA?PADWqWWkwLzGA?PeEsaI_h}MgWACBw@L_VcHQN?gACInDNP{kfGKTwACJqETaQOnvLJWACNEEtq?smsLYwACOHD^QU[j_LjgACR\EeQD?i\M`wAGA@@\?\CGrNGWAGIpDO@{{fGKSwAGKKD~aMknWKlgAGLmFRaAolzLiWAGP`DfAS_kXM@gAGRBEUqGci{MYGAKA[@E?XcGTMigAKJKCx@xGh?K[gAKLGDvqO[osK{WAKNHEtq?{mrLZWAKPDDWqWKkvLyWAKR@EUqGkizMXWAOA{@U?]{FtMpWAOJNCx@w{g~K\GAOLgEEaKooSLBGAOLmFSaAglwLiGAOOID^aU_j[LjgAOQ`ElaKWhDMQgASBV?~OZOHoMxWASJLCx@xCg}K[WASJqETQQ[nsLJGASMhFKaCgk|LRWASP_DfaS_k[MAWASQEE]qEkh_MIgAWBv@M?VcHON?GAWJJCx@xKg|K[wAWKLD}aMonXKlgAWNcFC`|OmWLbWAWOdDmQ[Kj@LrgAWPeEsQIki?MhWA[A@@[o\OGoNGGA[JMCx@x?g{K\WA[KlEMaSGmxKsWA[MHE|p}{l\LKgA[NiDuQYKj{MHGA[R`EdaD?iZM`WA_A\@EOXSGUMigA_JiDG@tWgaKcGA_LfEFAKooQLBwA_NbFD@|OmULawA_PcDeqS_kYM@wA_R_EeAD?i]M_wAcAx@T_^KFtMpgAcJhDG`tWg_KcwAcJqETAQSnuLIwAcMIE}@~?lXLKgAcOfDnA[Gj@LqgAcQ~EUqGsiyMXwAgBY?~OZWHnMwgAgJlDFptWg]KcWAgKMD}qM_nYKlgAgNFEtq?omvLZwAgNiDuAYCj}MIGAgQbElAKShDMQwAkBu@M_VcHMN?wAkJkDGPtWg`KbwAkKiEMASWmxKsgAkLmFSQAslyLhwAkPBDWqWSkuLywAkQFE\qEoh`MIgAoA@@[_\GGqNFwAoJjDF`tWg^KcgAoLJDvqOcorKzgAoMjFKACck|LRgAoOED^qUcj\LjgAoPeEtQIch|MhGAsA]@E_XWGQMigAsHqDV@z?gCKjgAsJqEUAQ_nwLIgAsLmFSAAklvLiwAsNiDtqYOjzMHwAsPeEtAI[h~MgwAwAz@T?^GFtMpwAwHqDUpywg@KjWAwKND~AMcnUKlgAwMgFKqC_k|LRwAwPEDWqW[kyLyGAwR^EdQD?i[M`gA{BW?~OZKHmMxGA{HqDU`zCgBKkWA{KkELaSSmxKswA{NfFD`|OmXLaWA{OFD_AUSj]LjgA{RAEUqGgixMYWB?Bt@LoVcHPN?WB?HqDV`y{g?KkGB?LHDvqOWoqK{GB?MJE|@~ClYLKgB?PbDfQS_kWMAgB?QdElqKchDMPwBCA@@\_\SGsNFgBCHqDVPysgAKjwBCLeEEQKooTLBWBCNDEtq?wmuLZGBCOcDma[Cj@LqwBCQBE]AEshaMIgBGBD@P?ZwG|MiGBGIyDIPxsgKKMGBGKuEHaPCo@KlGBGMtFFq@_mALKwBGOnDhqX?kGLjWBGQmEgAH[iHMJGBK@~@R_XkHiM}GBKIzDI`xcgLKMGBKJoEKAMwomL@GBKM~FR@|Gl^LTgBKOKDWQZskQMGGBKRZE^QIChBM[gBOBN@[OV_GYMqwBOIvDIpxggMKMGBOL?ESqKkn]KtwBONaE~AAGk{L]WBONhDkQUsktL~WBOQJEUaKOiRMfwBSAa?~_]kHFNEwBSIwDJ@xkgIKMGBSKRDwARwoJLHwBSLnFIP~SmnL_wBSP[D`AXgjAL{wBSQwErQDCheMRwBWBq@GO[_FvMzgBWIxDI@xogJKMGBWLbE?qPkmzK}gBWMQEuQCSmKLhgBWOxDtASgjdLsGBWPgEiaFOiuM^GB[BE@O?Z{G}MiGB[HsDKpvggxKaGB[JoEJqNCojL?wB[LnFI@~KmpL_gB[NhDkAUkkqL~GB[PgEiQFGiwM]wB_@~@ROXwHfM|wB_HsDK`v_gzK`wB_K~ESAKkn[KtWB_MOEuQC[mJLgwB_P]D_aXcjAL|GB_QnEgQHKiIMJGBcBM@Z_V_GWMqWBcHsDL`vWgwK`gBcKPDwAS?oILIWBcMpFGA@cmBLKwBcOwDtaSgjbLswBcRWE_AI?hBM[wBgA_?~_]sHENFWBgHsDLPvcgyK`WBgLdE?QPgmzK}wBgM}FQP|GlaLTGBgOoDiAXCkCLjWBgQHEUaKCiQMfGBkBs@Fo[[FvMzwBkHsDL@v[gvKaWBkKvEGaPGoAKlGBkN^E}aACk{L\WBkONDWQZgkPMGgBkQvErqDChcMRWBoBF@OO[?GyMiGBoJCDT`t[fhKUwBoK}ESaKkn^KuGBoM|FQp|Gl_LSgBoO{DsqSgj`LsWBoQuErADChfMSGBs@~@R?XoHhM|gBsJBDU@t[ffKUWBsKSDwARsoHLHgBsN`E~QA?k{L\gBsOpDiQWskDLjWBsPgEjQF?itM]gBwBL@[?V_GZMrGBwJADTPt[fiKVGBwLaD~qPwmzK~GBwLnFHp~WmmL_WBwOLDWQZokOMFwBwQjEgaHOiJMJGB{Ab?~_]gHDNEgB{J@DTpt[fgKUgB{KwEGqPKn}KlGB{MMEuQCOmNLhWB{NhDjqUwksM?GB{RYE^aH{hBMZwC?Bp@FO[kFvM{GC?JDDUPt[fjKUGC?JoEJaM{olL?gC?MqFFA@gmCLKwC?PZD`QX_jAL|WC?QKEUaKKiPMfgCCBB@O_[CGzMiGCCIVCwp{ggUKiwCCKQDwAR{oGLIGCCMPEuQCWmMLggCCOJDWQZwkNMGWCCQIEUaK?iTMgGCG@~@QoXgHjM}gCGITCwp{[gTKiGCGLcE?aPsmzK}GCGMrFFQ@WmDLKwCGNhDkqUokpL~wCGQtEraDChdMRgCKBP@ZOV_GXMqgCKIWCwp{cgSKigCKKsEHAPOn~KlGCKM{FRP|Gl]LTWCKP\D_qX[jAL|gCKPgEjAFKivM]WCOA`?~_]oHCNFGCOIUCwp{WgWKjGCOJoEJQMsonL@gCON]E}qAOk{L\wCOOzDtQSgjcLrwCOQkEgqHSiFMJGCSBr@G?[gFvMzGCSISCwp{_gVKiWCSLAERqKkn\KtgCSLnFH`~OmoL_GCSOqDhQWwkELjWCSR[E^AHwhBM[G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@om_[]oVkQmb]]vkPzDlQvgXym]pFkTzE_B_wS[GgFvkTzGjgfd|zG]qZSpyI]kfk`zH]qg@?NNFGy?XzH]rItTxb]rFkuuT]bfjXzK]rVky?OBs@qU_C]rVlAlS]Xvl@zPldfhHyy]sFlDzQ_DozWZogLflDzSj[FelzS]tZVlyU]fflPzT]tg@[MvE{iBPy|]tVlaln]Zvl`zXl^Fhhya]uFldzY_A@BKT{hjflqjq]Vvh\z[]vZ^lye]uflpz\]vg?_OsD]YYpz\]wInDxb]evm@z`l|vihy}]wFmDza_ApCoUlh]fmDzciuve\x~]xFmUwR]jfkHzc]xVmY?JCRPYid|]xVmajY]YvgLzg]yZ`Pyq]pfm`zh]yg@KO@DcyXXzh]zIqLxn]`vmpzllpvjXzI]zFmtzm_Cp?GUThdFmtzojGffLyJ]{FnEvG]mfkxzo]{VnI?UCE`[Qda]{VnQkK]\vg|zs]|Z\|y]]sfnPzt]|g@WOZDpiU@z\]|VnakJ]]vhLzw]}Z]@ya]tfn`zx]}n")

H5K4pnx = H5K4p.networkx_graph()

# List of graphs to process
graphs = [('H5K4p   ', H5K4p)]


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 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="")
                print(f"{label_i} IS isomorphic to {label_j}")
            else:
                print(f"{label_i} NOT isomorphic to {label_j}")
                
def non_isomorphic(graph_list):
    reps = []
    labels = []

    for label, G in graph_list:
        if not any(G.is_isomorphic(H) for _, H in reps):
            reps.append((label, G))
            labels.append(label.strip())   # remove extra spaces if you want    
    return labels

def isomorphic(graph_list):
    labels = []

    for i, (label, G) in enumerate(graph_list):
        if any(i != j and G.is_isomorphic(H) for j, (_, H) in enumerate(graph_list)):
            labels.append(label.strip())

    return labels

def compare_graphs_list(graphs):
    canon = {}
    count =1
    for label, G in graphs:
        s6 = G.canonical_label().sparse6_string()

#        print(f"{label.strip():15} : {s6[:40]}...")

        canon.setdefault(s6, []).append(label.strip())
    print("\n noniso ->",len(canon))
    print("\nGroups:")
    for group in canon.values():
        if len(group) > 1:
            print(count,"Isomorphic :", ", ".join(group))
            count=count+1
        else:
            print(count,"Unique      :", group[0])
            count=count+1



print("\n START \n")

#noniso = non_isomorphic(graphs)
#print(len(noniso), " non-isomorphic -> ",noniso)

#iso=isomorphic(graphs)
#print(len(iso), " isomorphic -> ",iso,"\n")


#compare_graphs_list(graphs)


nonisographs = graphs



# Print properties for each graph in the list
print("\n Main properties of the graph\n")
for label, graph in nonisographs:
    print(f"{label} | Ord.: {graph.order()} / Size: {graph.size()} / Diam.: {graph.diameter()} / Avg.dist: {graph.average_distance().n(digits=6)} / 6-reg.? {graph.is_regular(k=6)} /  Girth: {graph.girth()}   ")#/  Alg.conn. {algebraic_connectivity(graph).n(digits=6)} ")# /   Domin. number: {graph.dominating_set(value_only=True)} ")

print("\n Symmetry properties of the graph\n")
for label, graph in nonisographs:
    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()}" )

print("Degree histogram H5K4p   :", nx.degree_histogram(H5K4pnx) )  
    
'''    
print("\n Properties of the graphs as at arXiv\n")
for label, graph in nonisographs:
    print(f"{label} & {graph.average_distance().n(digits=6)}  & {graph.girth()}  &  {algebraic_connectivity(graph).n(digits=6)}   &  {graph.automorphism_group().order()} \\\\ ")
''' 
    
# Check isomorphisms
# print(f"Are isomorphic G5 and G6? {G5.is_isomorphic(G6)}")

#check_all_isomorphisms(nonisographs)

print("\n")

'''
# automorphism group structure
print(" Automorphism group structure")
print(' C_n is the cyclic group of order n;  x means direct product; : means semidirect product.\n')

AH5K4p   = H5K4p.automorphism_group()
print('H5K4p   :', AH5K4p.structure_description(), ' |  center order:',AH5K4p.center().order())

print("\n")
'''


# 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 nonisographs:
    print(f"{label} distance distrib from vtx. 0: {distance_distribution(graph, 0)}")


# Counting k-cycles for each graph
print("\nNumber of k-cycles for k=3 up to 4")
for label, graph in nonisographs:
    print(f"{label} ","  &  ".join(str(count_k_cycles(graph, k)) for k in range(3, 5)))




'''
# from 0 versio Molodtsov?

for name, G in nonisographs:
    fname = name.strip() + "_adjlst.txt"
    s = "; ".join(
        "{}-{}".format(v, ",".join(map(str, G.neighbors(v))))
        if G.neighbors(v) else str(v)
        for v in G.vertices(sort=True)
    )
    with open(fname, "w") as f:
        f.write(s)

'''
# from 0
       
for label, _ in nonisographs:
    base = label.strip()            
    G = globals()[base + "nx"]      

    with open(f"{base}_edges.txt", "w") as f:
        f.write(",".join(
            f"{{{u},{v}}}"
            for u, v in sorted((min(u, v), max(u, v)) for u, v in G.edges())
        ))

'''
# from 1
        
for label, _ in nonisographs:
    base = label.strip()   
    G = globals()[base + "nx"] 

    with open(f"{base}_Zedges.txt", "w") as f:
        f.write(",".join(
            f"{{{u+1},{v+1}}}"
            for u, v in sorted((min(u, v), max(u, v)) for u, v in G.edges())
        ))
'''       
print("\n DONE \n")        
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