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Algebraic Invariants and Some Hamiltonian Properties Graphs Rao Li Dept. of mathematical sciences

The 24th Clemson mini-Conference on Discrete Mathematics and Algorithms Oct. 22 – Oct. 23, 2009 Clemson University. Algebraic Invariants and Some Hamiltonian Properties Graphs Rao Li Dept. of mathematical sciences University of South Carolina Aiken Aiken, SC 29801. Outline

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Algebraic Invariants and Some Hamiltonian Properties Graphs Rao Li Dept. of mathematical sciences

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  1. The 24th Clemson mini-Conference on Discrete Mathematics and AlgorithmsOct. 22 – Oct. 23, 2009 Clemson University Algebraic Invariants and Some Hamiltonian Properties Graphs Rao Li Dept. of mathematical sciences University of South Carolina Aiken Aiken, SC 29801

  2. Outline -Some Results on Hamiltonian Properties of Graphs. -Algebraic Invariants. -Sufficient Conditions for Some Hamiltonian Properties of Graphs.

  3. 1. Some Hamiltonian Properties of Graphs. -A graph G is Hamiltonian if G has a Hamiltonian cycle, i.e., a cycle containing all the vertices of G. -A graph G is traceable if G has a Hamiltonian path, i.e., a path containing all the vertices of G. -A graph G is Hamiltonian-connected if there exists a Hamiltonian path between each pair of vertices in G.

  4. Dirac type conditions on Hamiltonian properties of graphs -Theorem 1. A graph G of order n is Hamiltonian if δ(G) ≥ n/2. -Theorem 2. A graph G of order n is traceable if δ(G) ≥ (n – 1)/2. -Theorem 3. A graph G of order n is Hamiltonian-connected if δ(G) ≥ (n + 1)/2.

  5. Ore type conditions on Hamiltonian properties of graphs -Theorem 4. A graph G of order n is Hamiltonian if d(u) + d(v) ≥ n for each pair of nonadjacent vertices u and v in G. -Theorem 5. A graph G of order n is traceable if d(u) + d(v) ≥ n – 1 for each pair of nonadjacent vertices u and v in G. -Theorem 6. A graph G of order n is Hamiltonian-connected if d(u) + d(v) ≥ n + 1 for each pair of nonadjacent vertices u and v in G.

  6. Closure theorems on Hamiltonian properties of graphs -The k - closure of a graph G, denoted clk(G), is a graph obtained from G by recursively joining two nonadjacent vertices such that their degree sum is at least k. -J. A. Bondy and V. Chvátal, A method in graph theory, Discrete Math. 15 (1976) 111-135.

  7. Closure theorems on Hamiltonian properties of graphs -Theorem 7. A graph G of order n has a Hamiltonian cycle if and only if cln(G) has one. -Theorem 8. A graph G of order n has a Hamiltonian path if and only if cln – 1(G) has one.

  8. Closure theorems on Hamiltonian properties of graphs -Theorem 9. A graph G of order n is Hamiltonian-connected if and only if cln + 1(G) is Hamiltonian-connected. -P. Wong, Hamiltonian-connected graphs and their strong closures, International J. Math. and Math. Sci. 4 (1997) 745-748.

  9. Closure theorems on Hamiltonian properties of graphs -Notice that every bipartite Hamiltonian graph must be balanced. -The k - closure of a balanced bipartite graph GBPT = (X, Y; E), where |X| = |Y|, denoted clk(GBPT), is a graph obtained from G by recursively joining two nonadjacent vertices x in X and y in Y such that their degree sum is at least k.

  10. Closure theorems on Hamiltonian properties of graphs -For a bipartite graph GBPT = (X, Y; E), define GCBPT = (X, Y; EC), where EC = { xy : x in E, y in E, and xy is not E }

  11. Closure theorems on Hamiltonian properties of graphs -Theorem 10. A balanced bipartite graph GBPT = (X, Y; E), where |X| = |Y| = r ≥ 2, has a Hamiltonian cycle if and only if clr + 1(GBPT) has one. -G. Hendry, Extending cycles in bipartite graphs, J. Combin. Theory (B) 51 (1991) 292-313.

  12. 2. Algebraic Invariants -The eigenvalues μ1(G) ≤ μ2(G) ≤ … ≤ μn(G) of a graph G are the eigenvalues of its adjacency matrix A(G). -The energy, denoted E(G), of a graph G is defined as |μ1(G)| + |μ2(G)| + … + |μn(G)|.

  13. -The Laplacian of a graph G is defined as L(G) = D(G) – A(G), where D(G) is the diagonal matrix of the vertex degrees of G. -The Laplacian eigenvalues 0 = λ1(G) ≤ λ2(G) ≤ … ≤ λn(G) of a graph G are the eigenvalues of L(G). -Σ2(G) := (λ1(G))2 + (λ2(G))2 + … + (λn(G))2 = sum of the diagonal entries in (L(G))2 = (d1(G))2 + d1(G) + (d2(G))2 + d2(G) … + (dn(G))2 + dn(G) = (d1(G))2 + (d2(G))2 + … + (dn(G))2 + 2e(G)

  14. 3. Sufficient Conditions for Some Hamiltonian Properties of Graphs -N. Fiedler and V. Nikiforov, Spectral radius and Hamiltonicity of graphs, to appear in Linear Algebra and its Applications. -Theorem 11. Let G be a graph of order n. [1] If μn(GC)≤ (n – 1)½, then G contains a Hamiltonian path unless G = Kn – 1 + v, a graph that consists of a complete graph of order n – 1 together with an insolated vertex v. [2] If μn(GC)≤ (n – 2)½, then G contains a Hamiltonian cycle unless G = Kn – 1+ e, a graph that consists of a complete graph of order n – 1 together with a pendent edge e.

  15. -Theorem 12. Let G be a 2-connceted graph of order n ≥ 12. [1] If μn(GC)≤ [(2n – 7)(n – 1)/n]½, then G contains a Hamiltonian cycle or G = Q2. [2] If Σ2(GC)≤ (2n – 7)(n + 1), then G contains a Hamiltonian cycle or G = Q2. where Q2 is a graph obtained by joining two vertices of the complete graph Kn – 2 to each of two independent vertices outside Kn – 2.

  16. in Theorem 12 Proof of [1] in Theorem 12.

  17. Proof of [1] in Theorem 12.

  18. Proof of [1] in Theorem 12.

  19. -Lemma 1. Let G be a 2-connceted graph of order n ≥ 12. If e(G) ≥ C(n – 2, 2) + 4, then G contains a Hamiltonian cycle or G = Q2. where C(n - 2, 2) = (n – 2)(n – 3)/2 and Q2 is a graph obtained by joining two vertices of the complete graph Kn – 2 to each of two independent vertices outside Kn – 2. -O. Byer and D. Smeltzer, Edge bounds in nonhamiltonian k-connected graphs, Discrete Math. 307 (2007) 1572-1579.

  20. 1 Proof of [1] in Theorem 12. Where K+2, n - 4 is defined as a graph obtained by joining the two vertices that are in the same color class of size two in K2, n - 4.

  21. in Theorem 12 Proof of [2] in Theorem 12.

  22. Proof of [2] in Theorem 12.

  23. From Lemma 2 below, we have that Proof of [2] in Theorem 12.

  24. -Lemma 2.Let X be a graph with n vertices and let Y be obtained from X by adding an edge joining two distinct vertices of X. Then λi(X) ≤ λi(Y), for all i, and λi(Y) ≤ λi+1(X), i < n. -Theorem 13.6.2, Page 291, C. Godsil and G. Royle, Algebraic Graph Theory, Springer Verlag, New York (2001).

  25. 1 Proof of [2] in Theorem 12. Lemma 2 again, we have that Where K+2, n - 4 is defined as a graph obtained by joining the two vertices that are in the same color class of size two in K2, n - 4.

  26. Other theorems on Hamiltonian properties of graphs -Theorem 13.Let G be a 3-connceted graph of order n ≥ 18. [1] If μn(GC) ≤ [3(n – 5)(n – 1)/n]½, then G contains a Hamiltonian cycle or G = Q3. [2] If Σ2(GC) ≤ 3(n – 5)(n + 1), then G contains a Hamiltonian cycle or G = Q3. Where Q3 is a graph obtained by joining three vertices of the complete graph Kn – 3 to each of three independent vertices outside Kn – 3.

  27. Other theorems on Hamiltonian properties of graphs -Theorem 14.Let G be a k-connceted graph of order n. [1] If μn(GC) ≤ [(kn – k2 + n – 2k - 3)(n – 1)/(2n)]½, then G contains a Hamiltonian cycle. [2] If Σ2(GC) ≤ (kn – k2 + n – 2k - 3)(n + 1)/2, then G contains a Hamiltonian cycle.

  28. Other theorems on Hamiltonian properties of graphs -Theorem 15.Let GBPT = (X, Y; E), where |X| = |Y| = r ≥ 2, be a balanced bipartite graph. [1] If μn(GBPTC) ≤ [(r – 2)/2]½, then GBPT contains a Hamiltonian cycle. [2] If Σ2(GBPTC) ≤ (r - 2)(r + 2), then GBPT contains a Hamiltonian cycle.

  29. Other theorems on Hamiltonian properties of graphs -Theorem 17.Let G be a graph of order n ≥ 7. [1] If μn(GC) ≤ [(n – 3)(n – 2)/n]½, then G is Hamiltonian-connected or G = Q. [2] If Σ2(GC) ≤ (n – 3)n, then G is Hamiltonian-connected or G = Q. Where Q is a graph obtained by joining two vertices of in the complete graph Kn – 1 to another vertex outside Kn – 1.

  30. Sufficient conditions involving energy for Hamiltonian properties of graphs -Theorem 18.Let G be a graph of order n ≥ 3. Then G contains a Hamiltonian cycle if [(n - 1)e(GC)/n]½((n + 1)½ + 1) + 2e(GC) – E(GC) < 2n – 4.

  31. -Lemma 3. Let e be any edge in a graph G. Then E(G) – 2 ≤ E(G – {e}) ≤ E(G) + 2. -J. Day and W. So, Singular value inequality and graph energy change, Electron. J. Linear Algebra 16 (2007) 291-299.

  32. in Theorem 18 Proof of Theorem 18.

  33. Proof of Theorem 18.

  34. Proof of Theorem 18.

  35. Proof of Theorem 18.

  36. Proof of Theorem 18.

  37. Proof of Theorem 18.

  38. Other sufficient conditions involving energy for Hamiltonian properties of graphs -Theorem 19.Let G be a graph of order n ≥ 2. Then G contains a Hamiltonian path if (e(GC))½((n - 1)½ + 1) + 2e(GC) – E(GC) < 2n – 2.

  39. Other sufficient conditions involving energy for Hamiltonian properties of graphs -Theorem 20.Let GBPT = (X, Y; E), where |X| = |Y| = r ≥ 2, be a balanced bipartite graph of order n = 2r ≥ 4. Then GBPT contains a Hamiltonian cycle if (e(GBPTC))½((n - 2)½ + 2½) + 2e(GBPTC) – E(GBPTC) < 2r – 2.

  40. Thanks

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