# SageMathCell online  https://sagecell.sagemath.org/?q=hobyjo
#  P'_11  Graph with 133 vertices and 792 edges, 12 vertices have degree 11 and 121 degree 12
#  deg. distrib.: [0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 121]
#  diameter: 2  avg. dist.:  1.90977
#  Girth: 3
# 

import networkx as nx

P11=Graph(r":~?AD_C?_C??S?_C??s??K?_?H_?D_?B_[A@k@@gM_wL`GL`OL__LACJ@kD@gP_oLASG@gS_WL`GM_wM_oM`_MBKD@sJ@oY`OMB[G@o\_OM_GN_OWCSDAW\CSEAgZCSKAoYCSHA?_COb_wVBGa__PBwa`OQBoaCcJA_[COd_W`COg__N_oNDkHB?XCgl_wTBwbDkCAo\DWl`WOBofDkKAwZD?lESGAG[C_l_gQBOkDgn`OSCGeDgs_W_DOlECH@{D@ww_wWBOcE_w_oRBobEow_OTDOrFCJAo_D?vFCKA?[DGtF?|`GVBgeE?w`?QCGdEOw__SBWkF?y_W^DWpF?z_wN`WNG[GB?ZDOpFpB__RBGeG@B`OTB_kEP@G[AAocDw|GXF_gVBwfEhAGXE`GPCGjExB`_QBgbEWzGXI_wSC?hEo{G[BBodE_~G[DB?[CWmG@K`ORBwgDo|HhM`?TC?eDpAG{AA?jDo~HCEAw`C_mFpI`_PBoiDp@HHP`WQBGhDoyHXO`GSBOfDpD_W\D_mF`EIkK@{I@xV_oWBghEO~GxV`?RBOjEg{HPNIxX_gTBGgEozGhPI{HAo]D_pH`MI{FA?`DOnGPOI{AAwdEwyHHSIx[`WPBWbE_|GpUIx\__QC?fGHLIPVJ[KA_^C_oG@GIhV__WBogFHII@``_RC?dDwxH`R_gUCGhEWxHhTJkGA?^D_sFHJIH[`OVBObEGxGxQJxb_OPCoqFHDHx]_wQB_jE?xIp_K{EA_XDOvFHEJHd_WZCwuFHGI`Y`WWBweDw{G`QK@d`GRBWcEXCI@[LCKAg`CwpFPCHxh`?UBGbE@@G`RJPc`OOBgdEo}G`UJpaLsCAw[DO~G`MJhi_oPC?kEgzG``Kxo_OQD?sGPCIhXK[BBOhEx?G`PJXeLcIB?_DWrFPEIHWK`p`WRCGkE?~GhSJ@b__TBOdFhJIhWKho_oUB_fE_{HHWKPrMkDA?ZCovGHIHpWMSGAw]DGnFXGIpWL@lNCFAG\D?pG@RJ@iL[AA_bEg}HhOJ@hL`y_WXC_qGPKIPWKxmMs@B?`Eo|HHRJHgL`u_GRB_vFXFIh]K`jNS@Ag]E?}GpQJ`iMPs_GUBwqFPIIpZKXoN[@A?XEG{HhSKHeMHwNk@AGYEXAH@MK@hLxt_GQBWtFxKIH^KPlN`}_GSBgnGHJHx\KxrMx~")
         
         
P11nx = P11.networkx_graph()

# List of graphs to process
graphs = [('P11   ', P11 )]


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 graph\n")
for label, graph in graphs:
    print(f"{label} | Ord.: {graph.order()} / Size: {graph.size()}  "
          f" / Diam.: {graph.diameter()} / Avg.dist: {graph.average_distance().n(digits=6)} \n"
          f" / 10-reg.? {graph.is_regular(k=10)} / Degree histogram: {nx.degree_histogram(P11nx)}  / Girth: {graph.girth()}\n "
          f" /  Alg.conn. {algebraic_connectivity(graph).n(digits=6)} /  Domin. number: {domination_number(graph)} ")

print("\n Symmetry properties of the graph\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()}" )


# 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 vtx. 0: {distance_distribution(graph, 0)}")

# Counting k-cycles for each graph
print("\nNumber of k-cycles for k=3 up to 5")
for label, graph in graphs:
    print(f"{label} ", " ".join(str(count_k_cycles(graph, k)) for k in range(3, 6)))

print("\n")
##