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2 laser
Published in E R Pike, High-power Gas Lasers, 1975, 2020
where gi is the degree of degeneracy. For the v2 mode gi = 2; for all other modes gi= 1. By writing a master kinetic equation for the transfer of energy between the various modes of the system, it is possible to obtain closed-form, albeit complex, equations for the time rate of energy change in a given vibrational mode. In the simplest case of V–T relaxation processes () dEidt≃1τ(E¯i−Ei)
Atom-Bond Connectivity Index
Published in Mihai V. Putz, New Frontiers in Nanochemistry, 2020
A descriptor is called degenerate if it possesses the same value for more than one graph. Dehmer et al. (2012a) evaluated the uniqueness, discrimination power or degeneracy of some degree, distance, and eigenvalue-based descriptors for investigating graphs holistically. In fact, the uniqueness of structural descriptors has been investigated in mathematical chemistry and related disciplines for discriminating the structure of isomeric structures and other chemical networks. A highly discriminating graph measure is desirable for analyzing graphs; hence, measuring the degree of its degeneracy is important for understanding its properties, limits, and quality. For the degree-based descriptors such as ABC, it is not surprising that these measures have only little discrimination power, as many graphs can be realized by identical degree sequences. This effect is even stronger if the cardinality of the underlying graph set increases. However, among the degree-based descriptors considered in Dehmer et al. (2012a), the ABC index had the highest discrimination power. In order to tackle the question of what kind of degeneracy the indices possess, Dehmer et al. (2012a) plotted their characteristic value distributions. For a graph class, they used the class of exhaustively generated non-isomorphic, connected and unweighted graphs with nine vertices. In the case of ABC, it was found that by using this index there exist many degenerate graphs possessing quite similar index values where the hull of the distributions forms a Gaussian curve.
Mesoscopic Systems
Published in James J Y Hsu, Nanocomputing, 2017
where the iqUantu is the principal quantum number, the index l is and azimuthal quantum number taking the values of 0, 1, ..., n – 1, and the index m is the magnetic quantum number. Up to the nth energy level, the degree of degeneracy is ∑1=0n-1(2l+1)=n2. $$ \mathop \sum \limits_{{1 = 0}}^{{n - 1}} (2l + 1) = n^{2} . $$
Distributed maximal independent set computation driven by finite-state dynamics
Published in International Journal of Parallel, Emergent and Distributed Systems, 2023
Eric Goles, Laura Leal, Pedro Montealegre, Ivan Rapaport, Martín Ríos-Wilson
A graph property is called an hereditary property if it is closed under taking induced subgraphs. An example of a hereditary property is bounded degeneracy. A graph G is called d-degenerate if every subgraph of G (including G itself) contains a vertex of degree at most d. Alternatively, a graph has degeneracy d if it can be decomposed successively removing vertices of degree at most d. In the following lemma, we now show that in a d-degenerate graph, for most vertices their degree is bounded by 4d−2.