Motivated by the necessities of the invariant theory of binary forms J. J. Sylvester constructed in 1878 for each graph with possible multiple edges but without loops its symmetrized graph monomial which is a polynomial in the vertex labels of the original graph. We pose the question for which graphs this polynomial is a non-negative resp. a sum of squares. This problem is motivated by a recent...

The requirement of a language to be conditionally decomposable is imposed on a specification language in the coordination supervisory control framework of discrete-event systems. In this paper, we present a polynomial-time algorithm for the verification whether a language is conditionally decomposable with respect to given alphabets. Moreover, we also present a polynomial-time algorithm to exte...

By Cayley’s theorem, any finite group G of order n can be regarded as a subgroup of the symmetric group $n. Let χ be any irreducible complex character of G and let V n χ (G) denote the symmetry classes of tensors associated with G and χ. In this paper assuming the Cayley representation of G, we obtain a formula for the dimension of V n χ (G) and discuss its non-vanishing in general. A necessary...

In this note, we symmetrized the cut-join equation from the proof of Marino-Vafa formula. One can derive more recursion formulas of Hodge integrals out of this polynomial equations. We also give some applications.

A n-vertex graph is said to be decomposable if for any partition (λ1, . . . , λp) of the integer n, there exists a sequence (V1, . . . , Vp) of connected vertex-disjoint subgraphs with |Vi| = λi. In this paper, we focus on decomposable trees. We show that a decomposable tree has degree at most 4. Moreover, each degree-4 vertex of a decomposable tree is adjacent to a leaf. This leads to a polyno...

Let K denote a field. A polynomial f(x) ∈ K[x] is said to be decomposable over K if f(x)= g(h(x)) for some polynomials g(x) and h(x)∈K[x] with 1< deg(h) < deg(f ). Otherwise f(x) is called indecomposable. If f(x)= g(xm) with m> 1, then f(x) is said to be trivially decomposable. In this paper, we show that xd+ax+b is indecomposable and that if e denotes the largest proper divisor of d, then xd+a...