# SageMathCell online https://sagecell.sagemath.org/?q=drkqly # (5,7) = 5516; Moore bound=27 306; # Communicated by Eyal Loz, Math Dep., Auckland Univ., New Zealand (July 2006) # https://web.archive.org/web/20091014041644/http://www.eyal.com.au/wiki/The_Degree/Diameter_Problem # ''' Main properties of the graph loz5516 | Ord.: 5516 / Size: 13790 / Diam.: 7 / Avg.dist: 5.86111 / 5-reg.? True / Girth: 7 Degree histogram : [0, 0, 0, 0, 0, 5516] Symmetry properties of the graph loz5516 | Aut.group.ord.: 2758 / Cayley ? False --- vtx.trans. ? False -- edge.trans. ? False Automorphism group structure: x means direct product; : means semidirect product. loz5516 : C2 x (C197 : C7) | center order: 2 loz5516 distance distrib from vtx. 0: [1, 5, 20, 80, 313, 1142, 2599, 1356] loz5516 distance distrib from vtx. 9: [1, 5, 20, 80, 315, 1142, 2611, 1342] ''' import networkx as nx loz5516 = 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loz5516nx =loz5516.networkx_graph() # List of graphs to process graphs = [('loz5516 ', loz5516)] 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() 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 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()} / Diam.: {graph.diameter()} / Avg.dist: {graph.average_distance().n(digits=6)} / 5-reg.? {graph.is_regular(k=5)} / Girth: {graph.girth()} ")# / Alg.conn. {algebraic_connectivity(graph).n(digits=6)} ")#/ Domin. number: {domination_number(graph)} ") print("Degree histogram :", nx.degree_histogram(loz5516nx)) 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()}" ) print("\n") ''' compare_graphs_list(graphs) ''' # automorphism group structure print(" Automorphism group structure: x means direct product; : means semidirect product.\n") Aloz5516 = loz5516.automorphism_group() print('loz5516 :', Aloz5516.structure_description(), ' | center order:',Aloz5516.center().order()) print("\n") ''' 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()} \\\\ ") ''' # 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 for label, graph in graphs: print(f"{label} distance distrib from vtx. 0: {distance_distribution(graph, 0)}") print(f"{label} distance distrib from vtx. 9: {distance_distribution(graph, 9)}") ''' # Counting k-cycles for each graph print("\nNumber of k-cycles for k=3 up to 7") for label, graph in graphs: print(f"{label} ", " ".join(str(count_k_cycles(graph, k)) for k in range(3, 8))) print("\n") # from 0 versio Molodtsov for name, G in graphs: 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 graphs: 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 graphs: 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") ##