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On the $A_{\alpha}$-spectrum of the $k$-splitting signed graph and neighbourhood coronas | ||
| Communications in Combinatorics and Optimization | ||
| مقاله 11، دوره 11، شماره 1، خرداد 2026، صفحه 155-169 اصل مقاله (481.06 K) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2024.29723.2133 | ||
| نویسندگان | ||
| Shariefuddin Pirzada* ؛ Mir Riyaz ul Rashid | ||
| Department of Mathematics, University of Kashmir, Srinagar, India | ||
| چکیده | ||
| Let $\Sigma=(G,\sigma)$ be a signed graph with adjacency matrix $A(\Sigma)$ and $D(G)$ be the diagonal matrix of its vertex degrees. For any real $\alpha\in [0,1]$, the $A_{\alpha}$-matrix of a signed graph $\Sigma$ is defined as $A_{\alpha}(\Sigma)=\alpha D(G)+(1-\alpha)A(\Sigma)$. Given a signed graph $\Sigma$ with vertex set $V=\{v_1, v_2,\dots, v_n\}$, the $k$-splitting signed graph $SP_k(\Sigma)$ of $\Sigma$ is obtained by adding to each vertex $v\in V(\Sigma)$ new $k$ vertices say $u^1, u^2, \ldots, u^k$ and joining every neighbour say $u$ of the vertex $v$ to $u^i$, $1\le i\le k$ by an edge which inherits the sign from $uv$. In this paper, we determine the $A_{\alpha}$-spectrum of $SP_k(\Sigma)$ in case of $\Sigma$ being a regular signed graph. For $k=1$, we introduce two distinct coronas of signed graphs $\Sigma_1$ and $\Sigma_2$ based on $SP_1(\Sigma_1)$, namely the splitting V-vertex neighbourhood corona and the splitting S-vertex neighbourhood corona. By examining the $A_{\alpha}$-characteristic polynomial of the resulting signed graphs, we derive their $A_{\alpha}$-spectra under certain regularity conditions on the constituent signed graphs. As applications, we use these results to construct infinite pairs of nonregular $A_{\alpha}$-cospectral signed graphs. | ||
| کلیدواژهها | ||
| signed graph؛ $k$-splitting signed graph؛ regular signed graph؛ $A_{\alpha}$-matrix؛ cospectrality | ||
| مراجع | ||
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[1] M. Basunia, I. Mahato, and M.R. Kannan, On the $A_{\alpha}$-spectra of some join graphs, Bull. Malays. Math. Sci. Soc. 44 (2021), no. 6, 4269–4297. https://doi.org/10.1007/s40840-021-01166-z
[2] F. Belardo, M. Brunetti, and A. Ciampella, On the multiplicity of $A_{\alpha}(\gamma)$-eigenvalue of signed graphs with pendant vertices, Discrete Math. 342 (2019), no. 8, 2223–2233. https://doi.org/10.1016/j.disc.2019.04.024
[3] D.M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs: Third Edition, Academic Press, 1995.
[4] S. Hameed, K. Paul, and K.A. Germina, On co-regular signed graphs, Australas. J. Combin. 62 (2015), no. 1, 8–17.
[5] S. Hamud and A. Berman, New constructions of nonregular cospectral graphs, Spec. Matrices 12 (2024), no. 1, Article ID: 20230109. https://doi.org/10.1515/spma-2023-0109
[6] S. Li and S. Wang, The $A_{\alpha}$-spectrum of graph product, Electron. J. Linear Algebra 35 (2019), 473–481. https://doi.org/10.13001/1081-3810.3857
[7] H. Lin, J. Xue, and J. Shu, On the $A_{\alpha}$-spectra of graphs, Linear Algebra Appl. 556 (2018), 210–219. https://doi.org/10.1016/j.laa.2018.07.003
[8] X. Liu and S. Liu, On the $A_{\alpha}$-characteristic polynomial of a graph, Linear Algebra Appl. 546 (2018), 274–288. https://doi.org/10.1016/j.laa.2018.02.014 [9] Z. Lu, X. Ma, and M. Zhang, Spectra of graph operations based on splitting graph, J. Appl. Anal. Comput. 13 (2023), no. 1, 133–155. https://doi.org/10.11948/20210446
[10] C. McLeman and E. McNicholas, Spectra of coronae, Linear Algebra Appl. 435 (2011), no. 5, 998–1007. https://doi.org/10.1016/j.laa.2011.02.007
[11] V. Nikiforov, Merging the A-and Q-spectral theories, Appl. Anal. Discrete Math. 11 (2017), no. 1, 81–107.
[12] V. Nikiforov, G. Pastén, O. Rojo, and R.L. Soto, On the $A_{\alpha}$-spectra of trees, Linear Algebra Appl. 520 (2017), 286–305. https://doi.org/10.1016/j.laa.2017.01.029
[13] V. Nikiforov and O. Rojo, A note on the positive semidefiniteness of $A_{\alpha}(g)$, Linear Algebra Appl. 519 (2017), 156–163. https://doi.org/10.1016/j.laa.2016.12.042
[14] G. Pastén, O. Rojo, and L. Medina, On the $A_{\alpha}$-eigenvalues of signed graphs, Mathematics 9 (2021), no. 16, Article ID: 1990. https://doi.org/10.3390/math9161990
[15] S. Pirzada, An Introduction to Graph Theory, Universities Press, Orient Blackswan, Hyderabad, 2012.
[16] M.A. Sahir and S.M.A. Nayeem, $A_{\alpha}$-spectra of graphs obtained by two corona operations and aα-cospectral graphs, Discrete Math. Algorithms Appl. 14 (2022), no. 2, Article ID: 2150112. https://doi.org/10.1142/S1793830921501123
[17] D. Sinha, P. Garg, and H. Saraswat, On the splitting signed graphs, J. Comb. Inf. Syst. Sci. 38 (2013), no. 1–4, Article ID: 103.
[18] Z. Stanić, A decomposition of signed graphs with two eigenvalues, Filomat 34 (2020), no. 6, 1949–1957. https://doi.org/10.2298/FIL2006949S
[19] Z. Stanić, On cospectral oriented graphs and cospectral signed graphs, Linear Multilinear Algebra 70 (2022), no. 19, 3689–3701. https://doi.org/10.1080/03081087.2020.1852153
[20] M.A. Tahir and X.D. Zhang, Coronae graphs and their $\alpha$-eigenvalues, Bull. Malays. Math. Sci. Soc. 43 (2020), no. 4, 2911–2927. https://doi.org/10.1007/s40840-019-00845-2
[21] E.R. Van Dam and W.H. Haemers, Which graphs are determined by their spectrum?, Linear Algebra Appl. 373 (2003), 241–272. https://doi.org/10.1016/S0024-3795(03)00483-X | ||
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