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A number field K is a finite extension of rational number field Q. A circulant digraph integral over K means that all its eigenvalues are algebraic integers of K. In this paper we give the necessary and sufficient condition for circulant digraphs which are integral over a number field K. We also solve Conjecture 3.3 in Xu and Meng (2011) [11] and find that it is affirmative.
Let φ:E→{1,2,…,k} be a proper edge coloring of a graph G=(V,E). The set of colors of edges incident to a vertex v∈V is called the color set of v and denoted by S(v). The coloring φ is vertex-distinguishing if S(u)≠S(v) for any two distinct vertices u,v∈V.A d-strong edge coloring of a graph G is a proper edge coloring that distinguishes any two distinct vertices u and v with distance 1≤d(u,v)≤d. The...
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