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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 LiDept. of mathematical sciences

University of South Carolina AikenAiken, SC 29801

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Outline

-Some Results on Hamiltonian Properties of Graphs.

-Algebraic Invariants.

-Sufficient Conditions for Some Hamiltonian

Properties of Graphs.

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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.

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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.

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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.

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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.

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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 an

d

only if cln – 1(G) has one.

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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.

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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.

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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 }

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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.

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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)|.

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-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)

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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.

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-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.

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Proof of [1] in Theorem 12.

in Theorem 12

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Proof of [1] in Theorem 12.

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Proof of [1] in Theorem 12.

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-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.

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Proof of [1] in Theorem 12.

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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.

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Proof of [2] in Theorem 12.

in Theorem 12

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Proof of [2] in Theorem 12.

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Proof of [2] in Theorem 12.

From Lemma 2 below, we have that

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-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).

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Proof of [2] in Theorem 12.

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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.

Lemma 2 again, we have that

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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.

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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.

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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.

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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.

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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.

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-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.

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Proof of Theorem 18.

in Theorem 18

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Proof of Theorem 18.

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Proof of Theorem 18.

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Proof of Theorem 18.

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Proof of Theorem 18.

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Proof of Theorem 18.

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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.

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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(GBPT

C) – E(GBPTC) < 2r – 2.

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Thanks