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On the Normalized Laplacian Permanental Polynomial of a Graph

  • Xiaogang LiuEmail author
  • Tingzeng Wu
Original Paper
  • 11 Downloads

Abstract

Let M be an \(n \times n\) matrix with entries \(m_{ij}\) (\(i,j=1,2,\ldots ,n\)). The permanent of M is defined to be
$$\begin{aligned} \mathrm{per}(M)=\sum _{\sigma }\prod _{i=1}^nm_{i\sigma (i)}, \end{aligned}$$
where the sum is taken over all permutations \(\sigma \) of \(\{1, 2, \ldots , n\}\). The permanental polynomial of M is defined by \(\mathrm{per}(xI_n-M),\) where \(I_n\) is the identity matrix of size n. Let G be a graph with n vertices. The normalized Laplacian matrix is defined to be \(\mathcal {L}(G)=D(G)^{-1/2}(D(G)-A(G))D(G)^{-1/2}\), where A(G) and D(G) denote the adjacency matrix and degree matrix of G, respectively. The permanental polynomial \(\mathrm{per}(xI_n-\mathcal {L}(G))\) is called the normalized Laplacian permanental polynomial of G. In this paper, we first give the formula to compute the coefficients of the normalized Laplacian permanental polynomial of a graph, and then we give recursive formulas to compute the normalized Laplacian permanental polynomial of a graph.

Keywords

Permanent Normalized Laplacian matrix Normalized Laplacian permanental polynomial 

Mathematics Subject Classification

05C31 05C50 15A15 

Notes

Acknowledgements

We greatly appreciate the anonymous referees for their comments and suggestions. We would also like to thank the referees for showing us useful references.

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Copyright information

© Iranian Mathematical Society 2019

Authors and Affiliations

  1. 1.Department of Applied MathematicsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  2. 2.Xi’an-Budapest Joint Research Center for CombinatoricsNorthwestern Polytechnical UniversityXi’anPeople’s Republic of China
  3. 3.School of Mathematics and StatisticsQinghai Nationalities UniversityXiningPeople’s Republic of China

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