Addendum to: “Hamilton-laceable bi-powers of locally finite bipartite graphs” [Discrete Math. 345 (2022) 112777]

In this addendum we give a short and easy negative answer to the two questions raised in the original article.     

Asymptotically Nonexpansive Mappings in Modular Function Spaces

In this paper, we prove that if ρ is a convex, σ-finite modular function satisfying a Δ₂-type condition, C a convex, ρ-bounded, ρ-a.e. compact subset of L_ρ, and T: C → C a ρ-asymptotically nonexpansive mapping, then T has a fixed point. In particular, any asymptotically nonexpansive self-map defined on a convex subset of L¹(Ω, μ) which is compact for the topology of local convergence in measure has a fixed point.     

On Clique-Transversals and Clique-Independent Sets

A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. A clique-independent set is a subset of pairwise disjoint cliques of G. Denote by _C (G) and _C (G) the cardinalities of the minimum clique-transversal and maximum clique-independent set of G, respectively. Say that G is clique-perfect when _C (H)= _C (H), for every induced subgraph H of G. In this paper, we prove that every graph not containing a 4-wheel nor a 3-fan as induced subgraphs and such that every odd cycle of length greater than 3 has a short chord is clique-perfect. The proof leads to polynomial time algorithms for finding the parameters _C (G) and _C (G), for graphs belonging to this class. In addition, we prove that to decide whether or not a given subset of vertices of a graph is a clique-transversal is Co-NP-Complete. The complexity of this problem has been mentioned as unknown in the literature. Finally, we describe a family of highly clique-imperfect graphs, that is, a family of graphs G whose difference _C (G)– _C (G) is arbitrarily large.     

The Flooding Time in Random Graphs

Based on our analysis of the hopcount of the shortest path between two arbitrary nodes in the class G _p (N) of random graphs, the corresponding flooding time is investigated. The flooding time T _N (p) is the minimum time needed to reach all other nodes from one node. We show that, after scaling, the flooding time T _N (p) converges in distribution to the two-fold convolution ^(2*) of the Gumbel distribution function (z)=exp (–e ^–z ), when the link density p _N satisfies Np _N /(log N)³ if N .     

Quantum Mayer Graphs for Coulomb Systems and the Analog of the Debye Potential

Within the Feynman–Kac path integral representation, the equilibrium quantities of a quantum plasma can be represented by Mayer graphs. The well known Coulomb divergencies that appear in these series are eliminated by partial resummations. In this paper, we propose a resummation scheme based on the introduction of a single effective potential that is the quantum analog of the Debye potential. A low density analysis of shows that it reduces, at short distances, to the bare Coulomb interaction between the charges (which is able to lead to bound states). At scale of the order of the Debye screening length ^–1 _D, approaches the classical Debye potential and, at large distances, it decays as a dipolar potential (this large distance behaviour is due to the quantum nature of the particles). The prototype graphs that result from the resummation obey the same diagrammatical rules as the classical graphs of the Abe–Meeron series. We give several applications that show the usefulness of to account for Coulombic effects at all distances in a coherent way.     

Adjoint L-Values and Primes of Congruence for Hilbert Modular Forms

Let f be a primitive Hilbert modular cusp form of arbitrary level and parallel weight k, defined over a totally real number field F. We define a finite set of primes that depends on the weight and level of f, the field F, and the torsion in the boundary cohomology groups of the Borel–Serre compactification of the underlying Hilbert-Blumenthal variety. We show that, outside , any prime that divides the algebraic part of the value at s=1 of the adjoint L-function of f is a congruence prime for f. In special cases we identify the boundary primes in terms of expressions of the form , where is a totally positive unit of F.     

关于Menger定理的推广的注

本文给出了一个图为Ｍｅｎｇｅｒ型的一个充分必要条件，利用这个条件，我们拓广了已知的Ｍｅｎｇｅｒ型图的类。     

<Emphasis Type="Italic">p</Emphasis>-Nuclearity in a New Perspective

In this Letter we try to settle some confused points concerning the use of the notion of p-nuclearity in the mathematical and physical literature, pointing out that the nuclearity index in the physicists’ sense vanishes for any p> 1. Our discussion of these issues suggests a new perspective, in terms of ε-entropy and operator spaces, which might permit connections to be drawn between phase space criteria and quantum energy inequalities.Mathematics Subject Classification (2000): 81T05, 47B10, 47L25.