scholarly journals Explicit estimates for Artin L-functions: Duke's short-sum theorem and Dedekind Zeta Residues

2021 ◽  
Author(s):  
Stephan Ramon Garcia ◽  
Ethan Simpson Lee
Keyword(s):  
Sum Theorem

scholarly journals Spectral equivalence of matrix polynomials and the Index Sum Theorem

2014 ◽  
Vol 459 ◽  
pp. 264-333 ◽  
Author(s):  
Fernando De Terán ◽  
Froilán M. Dopico ◽  
D. Steven Mackey
Keyword(s):  
Matrix Polynomials ◽  
Spectral Equivalence ◽  
Sum Theorem

scholarly journals Forcing $k$-Repetitions in Degree Sequences

10.37236/3503 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Yair Caro ◽  
Asaf Shapira ◽  
Raphael Yuster
Keyword(s):  
Graph Theory ◽  
Positive Integer ◽  
Main Tool ◽  
Geometric Approach ◽  
Degree Sequences ◽  
Integer Sequences ◽  
Sum Theorem ◽  
Zero Sum

One of the most basic results in graph theory states that every graph with at least two vertices has two vertices with the same degree. Since there are graphs without $3$ vertices of the same degree, it is natural to ask if for any fixed $k$, every graph $G$ is "close" to a graph $G'$ with  $k$ vertices of the same degree. Our main result in this paper is that this is indeed the case. Specifically, we show that for any positive integer $k$, there is a constant $C=C(k)$, so that given any graph $G$, one can remove from $G$ at most $C$ vertices and thus obtain a new graph $G'$ that contains at least $\min\{k,|G|-C\}$ vertices of the same degree.Our main tool is a multidimensional zero-sum theorem for integer sequences, which we prove using an old geometric approach of Alon and Berman.

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