JOURNAL OF SHANDONG UNIVERSITY(NATURAL SCIENCE) ›› 2023, Vol. 58 ›› Issue (11): 155-159.doi: 10.6040/j.issn.1671-9352.0.2022.095

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Two kinds of integral graphs based on complete graphs

Ligong WANG1(),Zhiming YU1,*(),Feng ZHOU2,Lijie TAO1,Luqi XING3   

  1. 1. School of Mathematics and Statistics, Northwestern Polytechnical University, Xi'an 710129, Shaanxi, China
    2. Honors College, Northwestern Polytechnical University, Xi'an 710129, Shaanxi, China
    3. School of Management, Northwestern Polytechnical University, Xi'an 710129, Shaanxi, China
  • Received:2022-02-07 Online:2023-11-20 Published:2023-11-07
  • Contact: Zhiming YU E-mail:lgwang@nwpu.edu.cn;yzm2020303246@mail.nwpu.edu.cn

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Abstract:

Two kinds of new graphs q * Kn and K1, m·(q * Kn) are defined. We obtain their characteristic polynomials by using the properties of characteristic polynomials of coalescence of graphs. We also characterize sufficient and necessary conditions with parameters for these graphs to be integral. Furthermore, some sufficient conditions for these graphs to be integral are presented.

Key words: integral graph, adjacency matrix, characteristic polynomial, complete graph

CLC Number: 

  • O157.5

Table 1

Parametric solutions in Corollary 3.2"

m n q a b c m n q a b c
5 5 2 5 1 -3 10 10 2 10 2 -4
10 4 3 5 1 -4 28 7 6 10 2 -7
28 4 9 8 1 -7 60 5 7 10 2 -9
21 5 10 9 1 -7 54 6 3 9 3 -8
33 5 16 11 1 -9 33 9 4 11 3 -7
55 4 18 11 1 -10 44 8 5 11 3 -8
20 5 2 6 2 -5 88 7 3 11 4 -10
28 5 3 7 2 -6 45 9 1 9 5 -7
16 7 3 8 2 -5 60 8 1 9 5 -8
24 6 4 8 2 -6 55 10 2 11 5 -8
1 HARARY F, SCHWENK A J. Which graphs have integral spectra[M]//Graphs and Combinatorics. Berlin: Springer, 1973: 18-22.
2 李学良, 林国宁. 关于整树问题[J]. 科学通报, 1987, 32 (11): 813- 816.
LI Xueliang , LIN Guoning . On integral trees problems[J]. Kexue Tongbao (Chinese), 1987, 32 (11): 813- 816.
3 BALIŃSKA K , CVETKOVIĆ D , RADOSAVLJEVIĆ Z , et al. A survey on integral graphs[J]. Publikacije Elektrotehnickog Fakulteta-Serija: Matematika, 2002, (13): 42- 65.
4 WANG Ligong. Integral trees and integral graphs[D]. Enschede: University of Twente, 2005.
5 CVETKOVIĆ D, ROWLINSON P, SIMIĆ S. An introduction to the theory of graph spectra[M]//New York: Cambridge University Press, 2010.
6 GHORBANI E , MOHAMMADIAN A , TAYFEH-REZAIE B . Integral trees of odd diameters[J]. Journal of Graph Theory, 2012, 70 (3): 332- 338.
doi: 10.1002/jgt.20619
7 WANG Ligong , WANG Qi , HUO Bofeng . Integral trees with diameter four[J]. Applied Mathematics and Computation, 2016, (282): 53- 64.
8 GHORBANI E , MOHAMMADIAN A , TAYFEH-REZAIE B . Integral trees with given nullity[J]. Discrete Mathematics, 2016, 339 (1): 157- 164.
doi: 10.1016/j.disc.2015.08.007
9 WANG Ligong , LI Xueliang , ZHANG Shenggui . Construction of integral graphs[J]. Applied Mathematics-A Journal of Chinese Universities, 2000, 15 (3): 239- 246.
doi: 10.1007/s11766-000-0046-z
10 BAPAT R B , KARIMI M . Integral complete multipartite graphs[J]. Linear Algebra and its Applications, 2018, 549, 1- 11.
doi: 10.1016/j.laa.2018.03.026
11 DE LIMA L S , MOHAMMADIAN A , OLIVEIRA C S . On integral graphs with at most two vertices of degree larger than two[J]. Linear Algebra and its Applications, 2020, 584, 164- 184.
doi: 10.1016/j.laa.2019.09.019
12 GORYAINOV S , KONSTANTINOVA E V , LI Honghai , et al. Integral graphs obtained by dual Seidel switching[J]. Linear Algebra and its Applications, 2020, 604, 476- 489.
doi: 10.1016/j.laa.2020.07.010
13 BRAGA R O , DEL-VECCHIO R R , RODRIGUES V M . Integral unicyclic graphs[J]. Linear Algebra and its Applications, 2021, 614, 281- 300.
doi: 10.1016/j.laa.2020.05.002
14 HUANG Jing , LI Shuchao . Integral and distance integral Cayley graphs over generalized dihedral groups[J]. Journal of Algebraic Combinatorics, 2021, 53 (4): 921- 943.
doi: 10.1007/s10801-020-00948-1
15 KONSTANTINOVA E V , LYTKINA D . Integral Cayley graphs over finite groups[J]. Algebra Colloquium, 2020, 27 (1): 131- 136.
doi: 10.1142/S1005386720000115
16 BAŠIĆ M . Characterization of strongly regular integral circulant graphs by spectral approach[J]. Applicable Analysis and Discrete Mathematics, 2022, 16 (2): 288- 306.
doi: 10.2298/AADM180713023B
17 CHENG Tao , FENG Lihua , LIU Weijun , et al. Distance powers of integral Cayley graphs over dihedral groups and dicyclic groups[J]. Linear and Multilinear Algebra, 2022, 70 (7): 1281- 1290.
doi: 10.1080/03081087.2020.1758609
18 LEPOVIĆ M . On integral graphs which belong to the class αGa∪βGb where Ga and Gbare two regular integral graphs[J]. Discrete Mathematics, 2023, 346 (5): 1- 11.
19 ARSIĆ B , CVETKOVIĆ D , SIMIĆ S K , et al. Graph spectral techniques in computer sciences[J]. Applicable Analysis and Discrete Mathematics, 2012, 6 (1): 1- 30.
doi: 10.2298/AADM111223025A
20 GODSIL C D , MCKAY B D . Constructing cospectral graphs[J]. Aequationes Mathematicae, 1982, 25 (1): 257- 268.
doi: 10.1007/BF02189621
21 BIGGS N . Algebraic graph theory[M]. New York: Cambridge University Press, 1993.
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