آمار
| تعداد نشریات | 6 |
| تعداد شمارهها | 118 |
| تعداد مقالات | 1,481 |
| تعداد مشاهده مقاله | 1,593,424 |
| تعداد دریافت فایل اصل مقاله | 1,490,441 |
On connected bipartite $Q$-integral graphs | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 2، دوره 10، شماره 4، اسفند 2025، صفحه 729-742 اصل مقاله (798.25 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29215.1895 | ||
| نویسندگان | ||
| Jesmina Pervin؛ Lavanya Selvaganesh* | ||
| Department of Mathematical Sciences, Indian Institute of Technology (Banaras Hindu University), Varanasi-221005, India | ||
| چکیده | ||
| A graph $G$ is said to be $H$-free if $G$ does not contain $H$ as an induced subgraph. Let $\mathcal{S}_{n}^2(m)$ be a \textit{variation of double star $\mathcal{S}_{n}^2$} obtained by adding m (<=n) disjoint edges between the pendant vertices which are at distance 3 in $\mathcal{S}_{n}^2$. A graph having integer eigenvalues for its signless Laplacian matrix is known as a Q-integral graph. The Q-spectral radius of a graph is the largest eigenvalue of its signless Laplacian. Any connected Q-integral graph G with Q-spectral radius 7 and maximum edge-degree 8 is either $K_{1,4}\square K_2$ or contains $\mathcal{S}_{4}^2(0)$ as an induced subgraph or is a bipartite graph having at least one of the induced subgraphs $\mathcal{S}_{4}^2(m)$, (m=1, 2, 3). In this article, we improve this result by showing that every connected Q-integral graph G having Q-spectral radius 7, maximum edge-degree 8 is always bipartite and $\mathcal{S}_{4}^2(3)$-free. | ||
| کلیدواژهها | ||
| Edge-degree؛ H-free graph؛ Signless Laplacian matrix؛ Q-integral graph | ||
| مراجع | ||
|
[1] A.E. Brouwer and W.H. Haemers, Spectra of graphs, Springer Science & Business Media, 2011.
[2] D. Cvetković and S.K. Simić, Towards a spectral theory of graphs based on the signless Laplacian, I, Publ. Inst. Math. 85 (2009), no. 105, 19–33. https://doi.org/10.2298/PIM0999019C
[3] 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
[4] M.A.A. de Freitas, N.M.M. de Abreu, R.R. Del-Vecchio, and S. Jurkiewicz, Infinite families of $Q$-integral graphs, Linear Algebra Appl. 432 (2010), no. 9, 2352–2360. https://doi.org/10.1016/j.laa.2009.06.029
[5] R.A. Horn and C.R. Johnson, Matrix Analysis, 2nd edition, Cambridge university press, 2013.
[6] A.F. Novanta, L. de Lima, and C.S. Oliveira, $Q$-integral graphs with at most two vertices of degree greater than or equal to three, Linear Algebra Appl. 614 (2021), 144–163. https://doi.org/10.1016/j.laa.2020.03.027
[7] 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
[8] 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
[9] 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
[10] S. Pirzada, On the conjecture for the sum of the largest signless Laplacian eigen-values of a graph-a survey, J. Discrete Math. Appl. 8 (2023), no. 3, 151–161. https://doi.org/10.22061/jdma.2023.10290.1061
[11] S. Pirzada, H.A. Ganie, and A.M. Alghamdi, On the sum of signless Laplacian spectra of graphs, Carpathian Math. Publ. 11 (2019), no. 2, 407–417. https://doi.org/10.15330/cmp.11.2.407-417
[12] 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 number: 152. https://doi.org/10.1007/s40314-023-02290-1
[13] 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
[14] 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 number: 26. https://doi.org/10.1007/s40314-023-02561-x
[15] 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
[16] S.K. Simić and Z. Stanić, On $Q$-integral $(3, s)$-semiregular bipartite graphs, Appl. Anal. Discrete Math. 4 (2010), no. 1, 167–174. http://doi.org/10.2298/AADM1000002S | ||
|
آمار تعداد مشاهده مقاله: 853 تعداد دریافت فایل اصل مقاله: 1,499 |
||