1 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

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 cl k (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)

7 Closure theorems on Hamiltonian properties of graphs -Theorem 7. A graph G of order n has a Hamiltonian cycle if and only if cl n (G) has one. -Theorem 8. A graph G of order n has a Hamiltonian path if and only if cl n – 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 cl n + 1 (G) is Hamiltonian-connected. -P. Wong, Hamiltonian-connected graphs and their strong closures, International J. Math. and Math. Sci. 4 (1997)

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 G BPT = (X, Y; E), where |X| = |Y|, denoted cl k (G BPT ), 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 G BPT = (X, Y; E), define G C BPT = (X, Y; E C ), where E C = { 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 G BPT = (X, Y; E), where |X| = |Y| = r ≥ 2, has a Hamiltonian cycle if and only if cl r + 1 (G BPT ) has one. -G. Hendry, Extending cycles in bipartite graphs, J. Combin. Theory (B) 51 (1991)

12 2. Algebraic Invariants -The eigenvalues μ 1 (G) ≤ μ 2 (G) ≤ … ≤ μ n (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 |μ 1 (G)| + |μ 2 (G)| + … + |μ n (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 = λ 1 (G) ≤ λ 2 (G) ≤ … ≤ λ n (G) of a graph G are the eigenvalues of L(G). -Σ 2 (G) := (λ 1 (G)) 2 + (λ 2 (G)) 2 + … + (λ n (G)) 2 = sum of the diagonal entries in (L(G)) 2 = (d 1 (G)) 2 + d 1 (G) + (d 2 (G)) 2 + d 2 (G) … + (d n (G)) 2 + d n (G) = (d 1 (G)) 2 + (d 2 (G)) 2 + … + (d n (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 μ n (G C ) ≤ (n – 1) ½, then G contains a Hamiltonian path unless G = K n – 1 + v, a graph that consists of a complete graph of order n – 1 together with an insolated vertex v. [2] If μ n (G C ) ≤ (n – 2) ½, then G contains a Hamiltonian cycle unless G = K n – 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 μ n (G C ) ≤ [(2n – 7)(n – 1)/n] ½, then G contains a Hamiltonian cycle or G = Q 2. [2] If Σ 2 (G C ) ≤ (2n – 7)(n + 1), then G contains a Hamiltonian cycle or G = Q 2. where Q 2 is a graph obtained by joining two vertices of the complete graph K n – 2 to each of two independent vertices outside K n – 2.

16 Proof of [1] in Theorem 12. 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 = Q 2. where C(n - 2, 2) = (n – 2)(n – 3)/2 and Q 2 is a graph obtained by joining two vertices of the complete graph K n – 2 to each of two independent vertices outside K n – 2. -O. Byer and D. Smeltzer, Edge bounds in nonhamiltonian k-connected graphs, Discrete Math. 307 (2007)

20 Proof of [1] in Theorem 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 K 2, n - 4.

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

22 Proof of [2] in Theorem 12.

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

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 λ i (X) ≤ λ i (Y), for all i, and λ i (Y) ≤ λ i+1 (X), i < n. -Theorem , Page 291, C. Godsil and G. Royle, Algebraic Graph Theory, Springer Verlag, New York (2001).

25 Proof of [2] in Theorem 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 K 2, n - 4. Lemma 2 again, we have that

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

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

28 Other theorems on Hamiltonian properties of graphs -Theorem 15. Let G BPT = (X, Y; E), where |X| = |Y| = r ≥ 2, be a balanced bipartite graph. [1] If μ n (G BPT C ) ≤ [(r – 2)/2] ½, then G BPT contains a Hamiltonian cycle. [2] If Σ 2 (G BPT C ) ≤ (r - 2)(r + 2), then G BPT 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 μ n (G C ) ≤ [(n – 3)(n – 2)/n] ½, then G is Hamiltonian-connected or G = Q. [2] If Σ 2 (G C ) ≤ (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 K n – 1 to another vertex outside K n – 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(G C )/n] ½ ((n + 1) ½ + 1) + 2e(G C ) – E(G C ) < 2n – 4.

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

32 Proof of Theorem 18. in 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(G C )) ½ ((n - 1) ½ + 1) + 2e(G C ) – E(G C ) < 2n – 2.

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

40 Thanks