Matchings in regular graphs from eigenvalues

Cioabă, Sebastian M.; Gregory, David A.; Haemers, Willem H.

doi:10.1016/j.jctb.2008.06.008

Cited by 94 publications

(46 citation statements)

References 14 publications

(21 reference statements)

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“…Cioabă and Gregory [] as well as Cioabă et al [] improved the above result. Furthermore, the following result obtained in [] is best possible. Theorem Let θ denote the greatest solution of

x^{3} + x^{2} - 6 x + 2 = 0

.…”

Section: Preliminariesmentioning

confidence: 85%

“…Then we have

τ = r

, and

δ (S, T) = k | S | + \sum_{x \in T} d_{G - S} (x) - k | T | - τ (S, T) = k m - r < 0 .

So by Theorem , G contains no k ‐factors. With the similar method in [], it is not difficult to prove that the third largest eigenvalue of this example is equal to the largest eigenvalue of

J (r, m)

, which equals

λ_{3} = λ_{1} (J (r, m)) = \frac{1}{2} (r - 2 + \sqrt{{(r + 2)}^{2} - 4 m}) < r - \frac{m - 1}{r + 1} + \frac{1}{(r + 1) (r + 2)} .

…”

Section: Main Theoremmentioning

confidence: 99%

“…S, T) = k|S|+ x∈T d G−S (x)−k|T|− (S, T) = km−r < 0.So by Theorem 1.6, G contains no k-factors. With the similar method in[4], it is not difficult to prove that the third largest eigenvalue of this example is equal to the largest eigenvalue of J(r, m), which equals 3 = 1(J(r, m)) = 1 2 (r −2+ (r +2) 2 −4m) < r − m−1 r +1 + 1 (r +1)(r +2).Journal of Graph Theory DOI 10.1002/jgt REGULAR GRAPHS, EIGENVALUES AND REGULAR FACTORS 355…”

mentioning

confidence: 99%

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Regular Graphs, Eigenvalues and Regular Factors

2011

Journal of Graph Theory

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“…Cioabă and Gregory [] as well as Cioabă et al [] improved the above result. Furthermore, the following result obtained in [] is best possible. Theorem Let θ denote the greatest solution of

x^{3} + x^{2} - 6 x + 2 = 0

.…”

Section: Preliminariesmentioning

confidence: 85%

“…Then we have

τ = r

, and

δ (S, T) = k | S | + \sum_{x \in T} d_{G - S} (x) - k | T | - τ (S, T) = k m - r < 0 .

So by Theorem , G contains no k ‐factors. With the similar method in [], it is not difficult to prove that the third largest eigenvalue of this example is equal to the largest eigenvalue of

J (r, m)

, which equals

λ_{3} = λ_{1} (J (r, m)) = \frac{1}{2} (r - 2 + \sqrt{{(r + 2)}^{2} - 4 m}) < r - \frac{m - 1}{r + 1} + \frac{1}{(r + 1) (r + 2)} .

…”

Section: Main Theoremmentioning

confidence: 99%

mentioning

confidence: 99%

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Regular Graphs, Eigenvalues and Regular Factors

2011

Journal of Graph Theory

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“…(b) If ≥ 4 is even, then is the intersection of the three closed half-spaces 1 , 3 , and 4 , and there are precisely two extreme points of , namely ( 1 ( + 1)…”

Section: Introductionmentioning

confidence: 99%

Tight lower bounds on the matching number in a graph with given maximum degree

Henning

Yeo

2018

Journal of Graph Theory

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Let k≥3. We prove the following three bounds for the matching number, α′false(Gfalse), of a graph, G, of order n size m and maximum degree at most k. If k is odd, then α′false(Gfalse)≥false(k−1kfalse(k2−3false)false)n+false(k2−k−2kfalse(k2−3false)false)m−k−1k(k2−3). If k is even, then α′false(Gfalse)≥nk(k+1)+mk+1−1k. If k is even, then α′false(Gfalse)≥false(k+2k2+k+2false)m−false(k−2k2+k+2false)n−k+2k2+k+2. In this article, we actually prove a slight strengthening of the above for which the bounds are tight for essentially all densities of graphs. The above three bounds are in fact powerful enough to give a complete description of the set Lk of pairs (γ,β) of real numbers with the following property. There exists a constant K such that α′false(Gfalse)≥γn+βm−K for every connected graph G with maximum degree at most k, where n and m denote the number of vertices and the number of edges, respectively, in G. We show that Lk is a convex set. Further, if k is odd, then Lk is the intersection of two closed half‐spaces, and there is exactly one extreme point of Lk, while if k is even, then Lk is the intersection of three closed half‐spaces, and there are precisely two extreme points of Lk.

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“…By using the Interlacing Theorem ( [2,5], Lemma 1.6 [4]) and the fact that the spectral radius of a graph is at least its average degree, they proved that if G is k-regular and has no perfect matching, then λ 3 (G) ≥ min i∈{1,2,3} λ 1 (H i ) > min H∈H 2|E(H)|/|V (H)|, where H i ∈ H. The bound on λ 3 that Brouwer and Haemers found could be improved, since equality in the bound on the spectral radius in terms of the average degree holds only when graphs are regular. Later, Cioabǎ, Gregory, and Haemers [3] found the minimum of λ 1 (H) over all graphs H ∈ H. More generally, Cioabǎ and the author [4] determined connections between the eigenvalues of an l-edge-connected k-regular graph and its matching number when 1 ≤ l ≤ k − 2.…”

Section: Introductionmentioning

confidence: 99%