A graph G is called spectrally d-degenerate if the largest eigenvalue of each subgraph of it with maximum degree D is at most √ dD. We prove that for every constant M there is a graph with minimum degree M which is spectrally 50-degenerate. This settles a problem of Dvořák and Mohar.

Abstract. We examine the periodic table of weak n-categories for the lowdimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to commutative monoids, and degenerate bicategories to monoidal categories; however, to understand this correspondence fully we examine the totalities of such structures together with maps between them ...

In this paper, we extend the notion of gapped strings to elastic-degenerate strings. An elastic-degenerate string can been seen as an ordered collection of k > 1 seeds (substrings/subpatterns) interleaved by elastic-degenerate symbols such that each elastic-degenerate symbol corresponds to a set of two or more variable length strings. Here, we present an algorithm for solving the pattern matchi...

A degenerate symbol x̃ over an alphabet Σ is a non-empty subset of Σ, and a sequence of such symbols is a degenerate string. A degenerate string is said to be conservative if its number of non-solid symbols is upper-bounded by a fixed positive constant k. We consider here the matching problem of conservative degenerate strings and present the first linear-time algorithm that can find, for given ...

In the study of quantum mechanical energy operators, particular attention is paid to the question of whether the bottom of the spectrum is an eigenvalue and whether that eigenvalue is simple. The corresponding eigenvector is usually called the gound state. Throughout our discussion we will say that a self-adjoint operator H has no ground state degeneracy when either E inf spectrum H is not an e...

This paper focuses on the geometric phase of general mixed states under unitary evolution. Here we analyze both non-degenerate as well as degenerate states. Starting with the non-degenerate case, we show that the usual procedure of subtracting the dynamical phase from the total phase to yield the geometric phase for pure states, does not hold for mixed states. To this end, we furnish an express...

In this paper, by introducing degenerate Fubini-type polynomials, with the help of Faà di Bruno formula and some properties partial Bell authors provide several new explicit formulas recurrence relations for polynomials numbers, associate newly defined Apostol–Bernoulli Apostol–Euler order α. These results enable one to present additional special numbers.

The property of degeneration of modular graded Lie algebras, first investigated by B. Weisfeiler, is analyzed. Transitive irreducible graded Lie algebras L = ∑ i∈Z Li, over algebraically closed fields of characteristic p > 2, with classical reductive component L0 are considered. We show that if a non-degenerate Lie algebra L contains a transitive degenerate subalgebra L′ such that dimL1 > 1, th...