Let $k$ be field of characteristic $p$, andlet $G$ be any finite group with splitting field $k$. Assume that $B$ is a $p$-block of $G$.In this paper, we introduce the notion of radical $B$-chain $C_{B}$, and we show that the $p$-local rank of $B$ is equals the length of $C_{B}$. Moreover, we prove that the vertex of a simple $kG$-module $S$ is radical if and only if it has the same vertex of th...

We initiate a general structure theory for vertex operator algebras V . We discuss the center and the blocks of V , the Jacobson radical and solvable radical, and local vertex operator algebras. The main consequence of our structure theory is that if V satisfies some mild conditions, then it is necessarily semilocal, i.e. a direct sum of local vertex operator algebras.

Each v ∈ V has a vertex operator Y (v, z) = ∑ n∈Z vnz −n−1 attached to it, where vn ∈ EndV. For the conformal vector ω we write Y (ω, z) = ∑ n∈Z L(n)z . If v is homogeneous of weight k, that is v ∈ Vk, then one knows that vn : Vm → Vm+k−n−1 and in particular the zero mode o(v) = vwtv−1 induces a linear operator on each Vm. We extend the “o” notation linearly to V, so that in general o(v) is the...

The radical of a vertex operator algebra associated to a module is defined and computed.

let z2 = {0, 1} and g = (v ,e) be a graph. a labeling f : v → z2 induces an edge labeling f* : e →z2 defined by f*(uv) = f(u).f (v). for i ε z2 let vf (i) = v(i) = card{v ε v : f(v) = i} and ef (i) = e(i) = {e ε e : f*(e) = i}. a labeling f is said to be vertex-friendly if | v(0) − v(1) |≤ 1. the vertex balance index set is defined by {| ef (0) − ef (1) | : f is vertex-friendly}. in this paper ...

Let R be a commutative ring with identity such that R admits at least two maximal ideals. In this article, we associate a graph with R whose vertex set is the set of all proper ideals I of R such that I is not contained in the Jacobson radical of R and distinct vertices I and J are joined by an edge if and only if I and J are not comparable under the inclusion relation. The aim of this article ...

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Given a polynomial φ(x) and a finite field Fq one can construct a directed graph where the vertices are the values in the finite field, and emanating from each vertex is an edge joining the vertex to its image under φ. When φ is a Chebyshev polynomial of prime degree, the graphs display an unusual degree of symmetry. In this paper we provide a complete description of these graphs, and then use ...

let g be a graph with p vertices and q edges and a = {0, 1, 2, . . . , [q/2]}. a vertex labeling f : v (g) → a induces an edge labeling f∗ defined by f∗(uv) = f(u) + f(v) for all edges uv. for a ∈ a, let vf (a) be the number of vertices v with f(v) = a. a graph g is said to be vertex equitable if there exists a vertex labeling f such that for all a and b in a, |vf (a) − vf (b)| ≤ 1 and the in...