Cayley subspace sum graph of vector spaces

Yıl 2023, Cilt: 33 Sayı: 33, 1 - 17, 09.01.2023

G. Kalaımurugan S. Gopınath T. Tamızh Chelvam

https://doi.org/10.24330/ieja.1195466

Öz

Let $\mathbb{V}$ be a finite dimensional vector space over the field $\mathbb{F}$. Let $S(\mathbb{V})$ be the set of all subspaces of $\mathbb{V}$ and $\mathbb{A}\subseteq S^*(\mathbb{V})=S(\mathbb{V})\backslash\{0\}.$ In this paper, we define the Cayley subspace sum graph of $\mathbb{V},$ denoted by Cay$(S^*(\mathbb{V}),\mathbb{A}), $ as the simple undirected graph with vertex set $S^*(\mathbb{V})$ and two distinct vertices $X$ and $Y$ are adjacent if $X+Z=Y$ or $Y+Z=X$ for some $Z\in \mathbb{A}$. Having defined the Cayley subspace sum graph, we study about the connectedness, diameter and girth of several classes of Cayley subspace sum graphs Cay$(S^*(\mathbb{V}), \mathbb{A})$ for a finite dimensional vector space $\mathbb{V}$ and $\mathbb{A}\subseteq S^*(\mathbb{V})=S(\mathbb{V})\backslash\{0\}.$

Anahtar Kelimeler

Cayley sum graph, vector space, subspace, diameter, girth, planar

Kaynakça

• M. Afkhami, K. Khashyarmanesh and K. Nafar, Generalized Cayley graphs associated to commutative rings, Linear Algebra Appl., 437(3) (2012), 1040-1049.
• M. Afkhami, M. R. Ahmadi, R. Jahani-Nezhad and K. Khashyarmanesh, Cayley graphs of ideals in a commutative ring, Bull. Malays. Math. Sci. Soc., 37(3) (2014), 833-843.
• M. Afkhami, Z. Barati, K. Khashyarmanesh and N. Paknejad, Cayley sum graphs of ideals of a commutative ring, J. Aust. Math. Soc., 96(3) (2014), 289-302.
• M. Afkhami, H. R. Barani, K. Khashyarmanesh and F. Rahbarnia, A new class of Cayley graphs, J. Algebra Appl. 15(4) (2016), 1650076 (8 pp).
• D. F. Anderson, T. Asir, A. Badawi and T. Tamizh Chelvam, Graphs from Rings, First ed., Springer, Cham, 2021.
• J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier Publishing Co., Inc., New York, 1976.
• P. J. Cameron, A. Das and H. K. Dey, On some properties of vector space based graphs, Linear Multilinear Algebra, https://doi.org/10.1080/03081087.2022.2121370, 2022.
• A. Das, Nonzero Component graph of a finite dimensional vector space, Comm. Algebra, 44(9) (2016), 3918-3926.
• A. Das, Subspace inclusion graph of a vector space, Comm. Algebra, 44(11) (2016), 4724-4731.
• A. Das, Non-zero component union graph of a finite-dimensional vector space, Linear Multilinear Algebra, 65(6) (2017), 1276-1287.
• A. Das, On subspace inclusion graph of a vector space, Linear Multilinear Algebra, 66(3) (2018), 554-564.
• A. V. Kelarev, On undirected Cayley graphs, Australas. J. Combin., 25 (2002), 73-78.
• C. Lanong and S. Dutta, Some results on graphs associated with vector spaces, J. Inf. Optim. Sci., 38(8) (2017), 1357-1368.
• T. Tamizh Chelvam and S. Anukumar Kathirvel, Generalized unit and unitary Cayley graphs of finite rings, J. Algebra Appl., 18(1)(2019), 1950006 (21 pp).
• T. Tamizh Chelvam and G. Kalaimurugan, Bounds for domination parameters in Cayley graphs on Dihedral group, Open J. Discrete Math., 2 (2012), 5-10.
• T. Tamizh Chelvam and K. Prabha Ananthi, The genus of graphs associated with vector spaces, J. Algebra Appl., 19(5) (2020), 2050086 (11 pp).
• T. Tamizh Chelvam, G. Kalaimurugan and W. Y. Chou, The signed star domination number of Cayley graphs, Discrete Math. Algorithms Appl., 4(2) (2012), 1250017 (10 pp).
• T. Tamizh Chelvam, K. Selvakumar and V. Ramanathan, Cayley sum graph of ideals of commutative rings, J. Algebra Appl., 17(7) (2018), 1850125 (14 pp).
• B. Tolue, Vector Space semi-Cayley graphs, Iran. J. Math. Sci. Inform., 13(2) (2018), 83-91.
• D. Wong, X. Wang and C. Xia, On two conjectures on the subspace inclusion graph of a vector space, J. Algebra Appl., 17(10)(2018), 1850189 (9 pp).