In this paper, some new properties are presented to the extremal graphs with largest (signless Laplacian) spectral radii in the set of all the connected graphs with prescribed degree sequences, via which we determine all the extremal tricyclic graphs in the class of connected tricyclic graphs with prescribed degree sequences, and we also prove some majorization theorems of tricyclic graphs with...

A sequence S is potentially Kp,1,1 graphical if it has a realization containing a Kp,1,1 as a subgraph, where Kp,1,1 is a complete 3-partite graph with partition sizes p, 1, 1. Let σ(Kp,1,1, n) denote the smallest degree sum such that every n-term graphical sequence S with σ(S) ≥ σ(Kp,1,1, n) is potentially Kp,1,1 graphical. In this paper, we prove that σ(Kp,1,1, n) ≥ 2[((p + 1)(n − 1) + 2)/2] ...

For any enumeration degree a let Ds a be the set of s-degrees contained in a. We answer an open question of Watson by showing that if a is a nontrivial Σ2-enumeration degree, then Ds a has no least element. We also show that every countable partial order embeds into Ds a. Finally, we construct Σ 0 2-sets A and B such that B ≤e A but for every X ≡e B, X s A.

A graph is called degree-magic if it admits a labelling of the edges by integers 1, 2, . . . , |E(G)| such that the sum of the labels of the edges incident with any vertex v is equal to 1+|E(G)| 2 deg(v). Degree-magic graphs extend supermagic regular graphs. In this paper we characterize complete tripartite degree-magic graphs.

As an extension of the Leray-Schauder degree, we introduce a topological degree theory for a class of demicontinuous operators of generalized (S+) type in real reflexive Banach spaces, based on the recent Berkovits degree. Using the degree theory, we show that the Borsuk theorem holds true for this class. Moreover, we study the Dirichlet boundary value problem involving the p-Laplacian by way o...

We propose an efficient way to obtain a correct Veneziano-Yankielowicz type integration constant of the effective glueball superpotential Weff (S, g,Λ), even for massless theories. Applying our method, we show some N = 1 theories do not have such an effective glueball superpotential, even though they have isolated vacua. In these cases, S = 0 typically. E-mail: [email protected]

Let G be a graph with (” : ‘) edges. We say G has an ascending subgraph decomposition (ASD) if the edge set of G can be partitioned into n sets generating graphs G,, G,, . , G, such that IE(G,)I = i (for i = 1, 2, . . , n) and G, is isomorphic to a subgraph of G,+r for i = 1,2, . . , n 1. In this note, we prove that if G is a graph of maximum degree d C [(n + 1)/2j on (” l ‘) edges, then G has ...

We show that a Boolean degree d function on the slice ([n] k ) = {(x1, . . . , xn) ∈ {0, 1} : ∑n i=1 xi = k} is a junta, assuming that k, n − k are large enough. This generalizes a classical result of Nisan and Szegedy on the hypercube. Moreover, we show that the maximum number of coordinates that a Boolean degree d function can depend on is the same on the slice and the hypercube.

Every hyperdegree at or above that of Kleene's O is the hyperjump and the supremum of two minimal hyperdegrees (Theorem 3.1). There is a nonempty ~ZX class of number-theoretic predicates each of which has minimal hyperdegree (Theorem 4.7). If V = L or a generic extension of L, then there are arbitrarily large hyperdegrees which are not minimal over any hyperdegree (Theorems 5.1, 5.2). If <?# ex...

There are various definitions for a Martin–Pour-El theory in the literature. We isolate two such definitions: weak Martin–Pour-El theories (which correspond to perfect thin Π1 classes) and strong Martin–Pour-El theories (which correspond to thin classes of separating sets). By concentrating on constructions of appropriate Π1 classes, rather than on direct constructions of the theories, we show ...