# SageMathCell online # https://sagecell.sagemath.org/?q=dcttny # Cayley graph with 2331 vertices and 11655 edges # Semidirect product Z_9 x(A) Z_259, A=44,81,123,256 # The graphs are non isomorphic # Z_40 x(44)Z_41 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# Z_40 x(81)Z_41 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# Z_40 x(123)Z_41 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# Z_40 x(256)Z_41 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# List of graphs to process graphs = [('Z_40 x(44)Z_41 ', G44), (' Z_40 x(81)Z_41', G81),('Z_40 x(123)Z_41 ', G123), (' Z_40 x(256)Z_41', G256)] def check_all_isomorphisms(graph_list): n = len(graph_list) print("\n Isomorphism check of all pairs (a dot means non 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(f"\n{label_i} is isomorphic to {label_j}") else: print(".", end="") 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()} / 10-reg.? {graph.is_regular(k=10)} " f" / Girth: {graph.girth()} / Diam.: {graph.diameter()} / Avg.dist: {graph.average_distance().n(digits=6)}") # / 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: {distance_distribution(graph, 0)}") ## Counting k-cycles for each graph #print("\nNumber of k-cycles for k=3 up to 10") #for label, graph in graphs: # print(f"{label} ", " ".join(str(count_k_cycles(graph, k)) for k in range(3, 11))) # #print("\n") ##