آمار
| تعداد نشریات | 5 |
| تعداد شماره‌ها | 117 |
| تعداد مقالات | 1,391 |
| تعداد مشاهده مقاله | 1,407,330 |
| تعداد دریافت فایل اصل مقاله | 1,374,387 |
On Connected Graphs with Integer-Valued Q-Spectral Radius | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 06 مهر 1403 اصل مقاله (500.51 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29278.1922 | ||
| نویسندگان | ||
| Jesmina Pervin؛ Lavanya Selvaganesh* ؛ Smrati Pandey | ||
| Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi-221005, India | ||
| چکیده | ||
| The $Q$-eigenvalues are the eigenvalues of the signless Laplacian matrix $Q(G)$ of a graph $G$, and the largest $Q$-eigenvalue is known as the $Q$-spectral radius $q(G)$ of $G$. The edge-degree of an edge is defined as the number of edges adjacent to it. In this article, we characterize the structure of simple connected graphs having integral $Q$-spectral radius. We show that the necessary and sufficient condition for such graphs to contain either a double star $\mathcal{S}_{r}^{2}$ or its variation $\mathcal{S}_{r}^{2,1}$ (having exactly one common neighbor between the central vertices) as a subgraph is that the maximum edge-degree is $2r$, where $r= q(G) -3$. In particular, we characterize all graphs that contain only double star as a subgraph when $q(G)$ equals $8$ and $9$. Further, we characterize all the connected edge-non-regular graphs with a maximum edge-degree equal to $4$ whose minimum $Q$-eigenvalue does not belong to the open interval $(0,1)$ and has an integral $Q$-spectral radius. | ||
| کلیدواژه‌ها | ||
| Edge-degree؛ Integral graph؛ Signless Laplacian matrix؛ $Q$-integral graph؛ $Q$-spectral radius | ||
| مراجع | ||
|
[1] C. Adiga, K. Das, and B.R. Rakshith, Some graphs determined by their signless Laplacian (distance) spectra, Electron. J. Linear Algebra 36 (2020), 461–472. https://doi.org/10.13001/ela.2020.4951 [2] K. Balinska, D. Cvetković, Z. Radosavljević, S. Simić, and D. Stevanović, A survey on integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. 13 (2002), 42–65.
[3] A.E. Brouwer and W.H. Haemers, Spectra of Graphs, Springer New York, 2011.
[4] D. Cvetković and S.K. Simić, Towards a spectral theory of graphs based on the signless Laplacian, II, Linear Algebra Appl. 432 (2010), no. 9, 2257–2272. https://doi.org/10.1016/j.laa.2009.05.020 [5] D.M. Cvetković, Cubic integral graphs, Univ. Beograd. Publ. Elektrotehn. Fak. (1975), no. 498/541, 107–113.
[6] A.H. Ghodrati and M.A. Hosseinzadeh, Signless Laplacian spectrum of a graph, Linear Algebra Appl. 682 (2024), 257–267. https://doi.org/10.1016/j.laa.2023.11.007 [7] F. Harary and A.J. Schwenk, Which graphs have integral spectra?, Graphs and Combinatorics (Berlin, Heidelberg) (R.A. Bari and F. Harary, eds.), Springer Berlin Heidelberg, 1974, pp. 45–51. https://doi.org/10.1007/BFb0066434 [8] R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 2013.
[9] I. Milovanović, E. Milovanović, M. Matejić, and Ş.B. Bozkurt Altındağ, Some remarks on the sum of the inverse values of the normalized signless Laplacian eigenvalues of graphs, Commun. Comb. Optim. 6 (2021), no. 2, 259–271. https://doi.org/10.22049/cco.2021.26987.1173 [10] S. Oh, J.R. Park, J. Park, and Y. Sano, On Q-integral graphs with Q-spectral radius 6, Linear Algebra Appl. 654 (2022), 267–288. https://doi.org/10.1016/j.laa.2022.08.031 [11] J. Park and Y. Sano, On Q-integral graphs with edge-degrees at most six, Linear Algebra Appl. 577 (2019), 384–411. https://doi.org/10.1016/j.laa.2019.04.015 [12] J. Pervin and L. Selvaganesh, Connected Q-integral graphs with maximum edge-degree less than or equal to 8, Discrete Math. 346 (2023), no. 3, Article ID: 113265. https://doi.org/10.1016/j.disc.2022.113265 [13] J. Pervin and L. Selvaganesh, On connected bipartite Q-integral graphs, Commun. Comb. Optim. (2024), In press. https://doi.org/10.22049/cco.2024.29215.1895 [14] S. Pirzada, On the conjecture for the sum of the largest signless Laplacian eigen-values of a graph-a survey, J. Disc. Math. Appl. 8 (2023), no. 4, 235–245. https://doi.org/10.22061/jdma.2023.10290.1061. [15] S. Pirzada and S. Khan, On Zagreb index, signless Laplacian eigenvalues and signless Laplacian energy of a graph, Comput. Appl. Math. 42 (2023), no. 4, Article ID: 152. https://doi.org/10.1007/s40314-023-02290-1 [16] S. Pirzada, B. Rather, R.U. Shaban, and T. Chishti, Signless Laplacian eigenvalues of the zero divisor graph associated to finite commutative ring $\mathbb{Z}_{p^{M_1}q^{M_2}}$, Commun. Comb. Optim. 8 (2023), no. 3, 561–574. https://doi.org/10.22049/cco.2022.27783.1353 [17] S. Pirzada, B. Rather, R.U. Shaban, and S. Merajuddin, On signless Laplacian spectrum of the zero divisor graphs of the ring Zn, Korean J. Math. 29 (2021), no. 1, 13–24. https://doi.org/10.11568/kjm.2021.29.1.13 [18] S. Pirzada, R.U. Shaban, H.A. Ganie, and L. de Lima, On the Ky Fan norm of the signless Laplacian matrix of a graph, Comput. Appl. Math. 43 (2024), no. 1, Article ID: 26. https://doi.org/10.1007/s40314-023-02561-x [19] S.K. Simić and Z. Stanić, Q-integral graphs with edge–degrees at most five, Discrete Math. 308 (2008), no. 20, 4625–4634. https://doi.org/10.1016/j.disc.2007.08.055. [20] Z. Stanić, There are exactly 172 connected Q-integral graphs up to 10 vertices., Novi Sad J. Math. 37 (2007), no. 2, 193–205.
[21] L. Xu and B. Zhou, Distribution of signless Laplacian eigenvalues and graph invariants, Linear Algebra Appl. 698 (2024), 589–602. https://doi.org/10.1016/j.laa.2024.06.019 [22] L. You, M. Yang, W. So, and W. Xi, On the spectrum of an equitable quotient matrix and its application, Linear Algebra Appl. 577 (2019), 21–40. https://doi.org/10.1016/j.laa.2019.04.013 | ||
|
آمار تعداد مشاهده مقاله: 226 تعداد دریافت فایل اصل مقاله: 209 |
||