A new method for proving chromatic uniqueness of graphs

Liu, Ruying; Zhao, Liang

doi:10.1016/s0012-365x(96)00078-7

Cited by 12 publications

(18 citation statements)

References 3 publications

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“…It is easy to see that all the chromatically unique graphs exhibited in [1,2,7] and many of chromatically unique graphs exhibited in [6,8,11] are special cases of this corollary.…”

Section: Corollary 33 For Any Graphs G With (G) > − 4 G Is Chromatmentioning

confidence: 86%

“…(3) Since h(P 5 ) = x 2 (x + 3)h(P 2 ). Then by Lemma 2.4(1),(8) and(10), we have (x + 3)|h(P i ), where i = 5, 11, 17, 23, 29. And, by Lemma 2.2 and h(P n+1 ) = x[h(P n ) + h(P n−1 )], we deduce that h(P 8 ) = h(P 2 )h(P 6 ) + xh(K 1 )h(P 5 ), h(P 14 ) = h(P 2 )h(P 12 ) + xh(K 1 )h(P 11 ), h(P 20 ) = h(P 2 )h(P 18 ) + xh(K 1 )h(P 17 ), h(P 26 ) = h(P 2 )h(P 24 ) + xh(K 1 )h(P 23 ).…”

mentioning

confidence: 91%

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The chromatic equivalence classes of the complements of graphs with the minimum real roots of their adjoint polynomials greater than -4

Ma¹,

Ren²

2008

Discrete Mathematics

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Let G be a graph with order n and G its complement. Denote by (G) the minimum real root of the adjoint polynomial of G. Two graphs G and H are chromatically equivalent if and only if G and H are adjointly equivalent. G is chromatically unique if and only if G is adjointly unique. In this paper, we give a method to determine all chromatic equivalence classes of a graph G with (G) > − 4, by using some results on the minimum real roots of the adjoint polynomial of G. Moreover, we obtain a necessary and sufficient condition for those graphs that are chromatically unique.

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“…It is easy to see that all the chromatically unique graphs exhibited in [1,2,7] and many of chromatically unique graphs exhibited in [6,8,11] are special cases of this corollary.…”

Section: Corollary 33 For Any Graphs G With (G) > − 4 G Is Chromatmentioning

confidence: 86%

mentioning

confidence: 91%

The chromatic equivalence classes of the complements of graphs with the minimum real roots of their adjoint polynomials greater than -4

Ma¹,

Ren²

2008

Discrete Mathematics

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“…It is clear that "∼" is an equivalence relation on the family of all graphs. By [G] we denote the equivalence class determined by G under [2][3][4], Liu [5], Liu and Zhao [6]). For a graph G with p vertices, the polynomial [8] gave all adjointly equivalent classes of P n .…”

Section: Introductionmentioning

confidence: 99%

On problems and conjectures on adjointly equivalent graphs

Zhao¹,

Li²,

Liu³

2005

Discrete Mathematics

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“…As a result, the adjoint polynomials of these two kinds of graphs S n and W n are obtained. Moreover, the adjoint polynomial of F n with n vertices is derived which a conjecture in [5] is addressed.…”

Section: Introductionmentioning

confidence: 99%

“…In [5], F 6 and F 10 were shown to be chromatically unique and the following conjecture was proposed.…”

Section: Introductionmentioning

confidence: 99%