The least Q-eigenvalue with fixed domination number

Yu, Guanglong; Zhai, Mingqing; Yan, Chao; Guo, Shu-Guang

doi:10.1016/j.amc.2018.07.055

Cited by 4 publications

(19 citation statements)

References 20 publications

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“…The q min among uncyclic graphs Lemma 4.1 [18] Let G be a nonbipartite unicyclic graph of order n and with the odd cycle C = v 1 v 2 · · · v g v 1 in it. There is a unit eigenvector X = ( x 1 , x 2 , .…”

Section: Note That There Is No Isolated Vertex Inmentioning

confidence: 99%

“…Lemma 4.2 [18] If G is a nonbipartite F • g,l -graph with g ≥ 5, n ≥ g + 1, then there is a graph H with girth 3 and order n such that γ(G) ≤ γ(H) and q min (H) < q min (G).…”

Section: Note That There Is No Isolated Vertex Inmentioning

confidence: 99%

“…., x n ) T is a unit eigenvector corresponding to q min (G). Then |x 3 | = max{|x 1 |, Lemma 4.5 [18] (i) If n = 4, the q K attains uniquely at H 3,1 ;…”

Section: Note That There Is No Isolated Vertex Inmentioning

confidence: 99%

“…A remaining open problem is that how about the q min of the connected nonbipartite graph on n vertices with domination number n+1 3 < γ ≤ n 2 . In [18], the authors proceeded on considering this problem. Among the nonbipartite graphs of order n = 4, the minimum q min is completely determined; among the nonbipartite graphs of order n and with given domination number n 2 , the minimum q min is completely determined; further results about the domination number, the q min of a graph as well as their relation are represented.…”

Section: Introductionmentioning

confidence: 99%

“…Let S = H 3,α be of order n ≥ 4 where α is the least integer such that ⌈ n−2α−2 3 ⌉ + α = γ. In [18], the authors represented some structural characterizations about the minimum q min for this problem, and conjectured that such S has the smallest q min . However, the problem seems really difficult to solve.…”

Section: Introductionmentioning

confidence: 99%

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Further results on the least Q-eigenvalue of a graph with fixed domination number

Zhai

2019

Linear and Multilinear Algebra

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In this paper, we proceed on determining the minimum q min among the connected nonbipartite graphs on n ≥ 5 vertices and with domination number n+1 3 < γ ≤ n−1 2 . Further results obtained are as follows:(i) among all nonbipartite connected graph of order n ≥ 5 and with domination number n−1 2 , the minimum q min is completely determined;(ii) among all nonbipartite graphs of order n ≥ 5, with odd-girth g o ≤ 5 and domination number at least n+1 3 < γ ≤ n−2 2 , the minimum q min is completely determined.

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Section: Note That There Is No Isolated Vertex Inmentioning

confidence: 99%

“…Lemma 4.2 [18] If G is a nonbipartite F • g,l -graph with g ≥ 5, n ≥ g + 1, then there is a graph H with girth 3 and order n such that γ(G) ≤ γ(H) and q min (H) < q min (G).…”

Section: Note That There Is No Isolated Vertex Inmentioning

confidence: 99%

“…., x n ) T is a unit eigenvector corresponding to q min (G). Then |x 3 | = max{|x 1 |, Lemma 4.5 [18] (i) If n = 4, the q K attains uniquely at H 3,1 ;…”

Section: Note That There Is No Isolated Vertex Inmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Further results on the least Q-eigenvalue of a graph with fixed domination number

Zhai

2019

Linear and Multilinear Algebra

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On the origins of enigmatic stellar populations in Local Group galactic nuclei

Leigh

Antonini

Stone

et al. 2016

Mon. Not. R. Astron. Soc.

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We consider the origins of enigmatic stellar populations in four Local Group galactic nuclei, specifically the Milky Way, M31, M32 and M33. These are centrally concentrated blue stars, found in three out of the four nuclear star clusters (NSCs) considered here. Their origins are unknown, but could include blue straggler (BS) stars, extended horizontal branch stars and young recently formed stars. Here, we calculate order-of-magnitude estimates for various collision rates, as a function of the host NSC environment and distance from the cluster centre. These rates are sufficiently high that BSs, formed via collisions between main sequence (MS) stars, could contribute non-negligibly (∼ 1-10% in mass) to every surface brightness profile, with the exception of the Milky Way. Stellar evolution models show that the envelopes of red giant branch (RGB) stars must be nearly completely stripped to significantly affect their photometric appearance, which requires multiple collisions. Hence, the collision rates for individual RGB stars are only sufficiently high in the inner 0.1 pc of M31 and M32 for RGB destruction to occur. Collisions between white dwarfs and MS stars, which should ablate the stars, could offer a steady and significant supply of gas in every NSC in our sample. The gas could either fragment to form new stars, or accrete onto old MS stars already present. Thus, collisional processes could contribute significantly to the observed blue excesses in M31 and M33; future studies should be aimed at better constraining theoretical predictions to compliment existing and future observational data.

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Small-N collisional dynamics – IV. Order in the realm of not-so-small-N

Leigh

Geller

Shara

et al. 2018

Monthly Notices of the Royal Astronomical Society

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In this paper, the fourth in the series, we continue our study of combinatorics in chaotic Newtonian dynamics. We focus once again on the chaotic four-body problem in Newtonian gravity assuming finite-sized particles, and interactions that produce direct collisions between any two particles. Our long-term goal is to predict the probability of a given collision event occurring over the course of an interaction, as a function of the numbers and properties of the particles. In previous papers, we varied the number of interacting particles, as well as the distributions of particle radii and masses. Here, we refine the methods developed in these preceding studies, and arrive at a final and robust methodology that can be used to study collisional dynamics in a variety of astrophysical contexts, ranging from stars in star clusters, galaxies in galaxy groups and clusters and even the collisional growth of planetesimals in protoplanetary disks. We further present and refine the concept of a Collision Rate Diagram (CRD), the primary tool we use to quantify the relative rates for different collision scenarios to occur. The agreement between our final theoretical model and the results of numerical scattering simulations is excellent.

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The least Q-eigenvalue with fixed domination number

Cited by 4 publications

References 20 publications

Further results on the least Q-eigenvalue of a graph with fixed domination number

Further results on the least Q-eigenvalue of a graph with fixed domination number

On the origins of enigmatic stellar populations in Local Group galactic nuclei

Small-N collisional dynamics – IV. Order in the realm of not-so-small-N

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