Related papers: The Degree Polynomial
We study the probability distribution of the number of zeros of multivariable polynomials with bounded degree over a finite field. We find the probability generating function for each set of bounded degree polynomials. In particular, in the…
Given an integer $\Delta \ge 3$, let ${\cal G}_{\Delta }$ be the set of connected graphs $G\neq K_{\Delta +1}$ with maximum degree $\Delta $ and, for $i=1,\cdots, \Delta $, let $V_i(G)$ be the set of vertices of $G$ of degree $i$. \\ We…
The neighborhood polynomial of graph $G$, denoted by $N(G,x)$, is the generating function for the number of vertex subsets of $G$ which are subsets of open neighborhoods of vertices in $G$. For any graph polynomial, it can be useful to…
Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of…
Let G be a finite group and A a finite dimensional G-graded algebra over a field of characteristic zero. When A is simple as a G-graded algebra, by mean of Regev central polynomials we construct multialternating graded polynomials of…
Let G(V,E) be a simple graph and let X subset of V. Two vertices u and v are said to be X-visible if there exists a shortest u,v-path P such that V(P) intersection X is a subset of {u, v}. A set X is called a mutual-visibility set of G if…
A "tree-partition" of a graph $G$ is a partition of $V(G)$ such that identifying the vertices in each part gives a tree. It is known that every graph with treewidth $k$ and maximum degree $\Delta$ has a tree-partition with parts of size…
The alliance polynomial of a graph $\Gamma$ with order $n$ and maximum degree $\delta_1$ is the polynomial $A(\Gamma; x) = \sum_{k=-\delta_1}^{\delta_1} A_{k}(\Gamma) \, x^{n+k}$, where $A_{k}(\Gamma)$ is the number of exact defensive…
We study polynomial iterative roots of polynomials and describe the locus of complex polynomials of degree 4 admitting a polynomial iterative square root.
We consider properties of polynomials with coefficients in division rings. A theorem on the decomposition of a polynomial with coefficients in an arbitrary division ring is obtained. It is shown that if a non-central element is not a root…
Each graph and choice of a commutative ring gives rise to an associated graphical group. In this article, we introduce and investigate graph polynomials that enumerate conjugacy classes of graphical groups over finite fields according to…
Let $G = (V, E)$ be a simple graph of order $n$. The total dominating set of $G$ is a subset $D$ of $V$ that every vertex of $V$ is adjacent to some vertices of $D$. The total domination number of $G$ is equal to minimum cardinality of…
We introduce and study the problem of optimizing arbitrary functions over degree sequences of hypergraphs and multihypergraphs. We show that over multihypergraphs the problem can be solved in polynomial time. For hypergraphs, we show that…
We introduce a new graph invariant \Lambda(G) that we call maxmaxflow, and put it in the context of some other well-known graph invariants, notably maximum degree and its relatives. We prove the equivalence of two "dual" definitions of…
For a simple graph $G$, let $\chi(G,x)$ denote the chromatic polynomial of $G$. This manuscript introduces some polynomials which are related to chromatic polynomial and their relations.
Let $F$ be a non-negatively graded free module over a polynomial ring $\mathbb{K}[x_1,\dots,x_n]$ generated by $m$ basis elements. Let $M$ be a submodule of $F$ generated by elements in $F$ with degrees bounded by $D$ and dim $F/M$=$r$. We…
The degree of a polynomial representing (or approximating) a function f is a lower bound for the number of quantum queries needed to compute f. This observation has been a source of many lower bounds on quantum algorithms. It has been an…
Suppose a finite, unweighted, combinatorial graph $G = (V,E)$ is the union of several (degree-)regular graphs which are then additionally connected with a few additional edges. $G$ will then have only a small number of vertices $v \in V$…
Working over the split octonions over an algebraically closed field, we solve all polynomial equations in which all the coefficients but the constant term are scalar. As a consequence, we calculate the n-th roots of an octonion.
Elementary Algebraic Geometry can be described as study of zeros of polynomials with integer degrees, this idea can be naturally carried over to `polynomials' with rational degree. This paper explores affine varieties, tangent space and…