1. On the Roots of (Signless) Laplacian Permanental Polynomials of Graphs.
- Author
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Wu, Tingzeng, Zeng, Xiaolin, and Lü, Huazhong
- Subjects
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LAPLACIAN matrices , *BIPARTITE graphs , *POLYNOMIALS , *LINEAR algebra , *MULTIPLICITY (Mathematics) - Abstract
Let G be a graph, and let Q(G) and L(G) denote the signless Laplacian matrix and the Laplacian matrix of G, respectively. The polynomials ϕ (Q (G) , x) = per (x I n - Q (G)) and ϕ (L (G) , x) = per (x I n - L (G)) are called signless Laplacian permanental polynomial and Laplacian permanental polynomial of G, respectively. In this paper, we investigate the properties of roots of ϕ (Q (G) , x) . We obtain the real root distribution of ϕ (Q (G) , x) . In particular, using the Gallai–Edmonds structure theorem, we determine the structures of graphs G whose roots of signless Laplacian permanental polynomial of G contain no positive integer p, where p is the minimum vertex degree of G. And we determine completely the graphs each of which having the multiplicity of the integer root p is equal to the deficiency of a maximum p-pendant structure of the graph. These results extend the conclusion obtained by Faria (Linear Algebra Appl 299:15–35, 1995). Furthermore, we give an algorithm to calculate the multiplicity of the root p of ϕ (Q (G) , x) . And we also determine the relation between the multiplicity of the root p of ϕ (Q (G) , x) and the matching number of G. Finally, we investigate the properties of roots of Laplacian permanental polynomial of non-bipartite graphs. And some open problems are presented. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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