1. The generalized distance matrix of digraphs.
- Author
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Xi, Weige, So, Wasin, and Wang, Ligong
- Subjects
- *
LAPLACIAN matrices , *SPECTRAL theory , *DISTANCES , *MATRICES (Mathematics) , *EUCLIDEAN distance , *RADIUS (Geometry) - Abstract
Let D (G) and D Q (G) = D i a g (T r) + D (G) be the distance matrix and distance signless Laplacian matrix of a simple strongly connected digraph G , respectively, where D i a g (T r) = diag (D 1 , D 2 , ... , D n) be the diagonal matrix with vertex transmissions of the digraph G. To track the gradual change of D (G) into D Q (G) , in this paper, we propose to study the convex combinations of D (G) and D i a g (T r) defined by D α (G) = α D i a g (T r) + (1 − α) D (G) , 0 ≤ α ≤ 1. This study reduces to merging the distance spectral and distance signless Laplacian spectral theories. The eigenvalue with the largest modulus of D α (G) is called the D α spectral radius of G , denoted by μ α (G). We determine the digraph which attains the maximum (or minimum) D α spectral radius among all strongly connected digraphs. Moreover, we also determine the digraphs which attain the minimum D α spectral radius among all strongly connected digraphs with given parameters such as dichromatic number, vertex connectivity or arc connectivity. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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