JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (12): 151-160.doi: 10.6040/j.issn.1671-9352.0.2022.316

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Characterizing properties of (signless) Laplacian permanental polynomials of bicyclic graphs

Tingzeng WU*(),Tian ZHOU   

  1. School of Mathematics, Qinghai Minzu University, Xining 810007, Qinghai, China
  • Received:2022-05-17 Online:2023-12-20 Published:2023-12-19
  • Contact: Tingzeng WU E-mail:mathtzwu@163.com

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Abstract:

Let G be a graph with n vertices, and let L(G) and Q(G) be the Laplacian matrix and signless Laplacian matrix of G, respectively. The polynomial π(L(G); x)=per(xI-L(G)) (resp. π(Q(G); x)=per(xI-Q(G))) is called Laplacian permanental polynomial (resp. signless Laplacian permanental polynomial) of G. In this paper, we show that two classes of bicyclic graphs are determined by their (signless) Laplacian permanental polynomials.

Key words: permanent, (signless) Laplacian matrix, (signless) Laplacian permanental polynomial, (signless) Laplacian copermanental

CLC Number: 

  • O157.6

Fig.1

Bycyclic graps d(p, q, r) and θ(p, q, r)"

1 VALIANT L G . The complexity of computing the permanent[J]. Theoretical Computer Science, 1979, 8 (2): 189- 201.
doi: 10.1016/0304-3975(79)90044-6
2 MERRIS R , REBMAN K R , WATKINS W . Permanental polynomials of graphs[J]. Linear Algebra and Its Applications, 1981, 38, 273- 288.
doi: 10.1016/0024-3795(81)90026-4
3 BAPAT R B . A bound for the permanent of the Laplacian matrix[J]. Linear Algebra and Its Applications, 1986, 74, 219- 223.
doi: 10.1016/0024-3795(86)90124-2
4 CASH G , GUTMAN I . The lapacian permanental polynomial: formulas and algorithms[J]. MATCH Communications in Mathematical and in Computer Chemistry, 2004, 51 (51): 129- 136.
5 GENG X Y , HU X , LI S C . Further results on permanental bounds for the Laplacian matrix of trees[J]. Linear and Multilinear Algebra, 2010, 58 (5): 571- 587.
doi: 10.1080/03081080902765583
6 GENG X Y , HU S N , LI S C . Permanental bounds of the Laplacian matrix of trees with given domination number[J]. Graphs and Combinatorics, 2015, 31 (5): 1423- 1436.
doi: 10.1007/s00373-014-1451-z
7 LI S C , LI Y , ZHANG X X . Edge-grafting theorems on permanents of Laplacian matrices of graphs and their applications[J]. The Electronic Journal of Linear Algebra, 2013, 26, 28- 48.
8 LIU S Y . On the (signless) Laplacian permanental polynomials of graphs[J]. Graphs and Combinatorics, 2019, 35 (3): 787- 803.
doi: 10.1007/s00373-019-02033-2
9 LIU X G , WU T Z . Computing the permanental polynomials of graphs[J]. Applied Mathematics and Computation, 2017, 304 (C): 103- 113.
10 MERRIS R . The Laplacian permanental polynomial for trees[J]. Czechoslovak Mathematical Journal, 1982, 32 (3): 397- 403.
doi: 10.21136/CMJ.1982.101816
11 VRBA A . Principal subpermanents of the Laplacian matrix[J]. Linear and Multilinear Algebra, 1986, 19 (4): 335- 346.
doi: 10.1080/03081088608817728
12 GOLDWASSER J L . Permanent of the Laplacian matrix of trees with a given matching[J]. Discrete Mathematics, 1986, 61 (2/3): 197- 212.
13 BOTTI P , MERRIS R , VEGA C . Laplacian permanents of trees[J]. SIAM Journal on Discrete Mathematics, 1992, 5 (4): 460- 466.
doi: 10.1137/0405036
14 LIU X G , WU T Z . Graphs determined by the (signless) Laplacian permanental polynomial[J]. Linear and Multilinear Algebra, 2017, 3599- 3615.
15 FARIA I . Permanental roots and the star degree of a graph[J]. Linear Algebra and Its Applications, 1985, 64, 255- 265.
doi: 10.1016/0024-3795(85)90281-2
16 FARIA I . Multiplicity of integer roots of polynomials of graphs[J]. Linear Algebra and Its Applications, 1995, 229, 15- 35.
doi: 10.1016/0024-3795(93)00337-Y
17 LI S C , ZHANG L . Permanental bounds for the signless Laplacian matrix of bipartite graphs and unicyclic graphs[J]. Linear and Multilinear Algebra, 2011, 59 (2): 145- 158.
doi: 10.1080/03081080903261467
18 LI S C , ZHANG L . Permanental bounds for the signless Laplacian matrix of a unicyclic graph with diameter d[J]. Graphs and Combinatorics, 2012, 28 (4): 531- 546.
doi: 10.1007/s00373-011-1057-7
19 WU T Z , ZHOU T , LV H Z . Further results on the star degree of graphs[J]. Applied Mathematics and Computation, 2022, 425, 127076.
doi: 10.1016/j.amc.2022.127076
20 BRUALDI R A , GOLDWASSER J L . Permanent of the Laplacian matrix of trees and bipartite graphs[J]. Discrete Mathematics, 1984, 48 (1): 1- 21.
doi: 10.1016/0012-365X(84)90127-4
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