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On graphs with anti-reciprocal eigenvalue property | ||
| Transactions on Combinatorics | ||
| دوره 13، شماره 1، خرداد 2024، صفحه 17-30 اصل مقاله (514.62 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2022.135210.2015 | ||
| نویسندگان | ||
| Sadia Akhter؛ Uzma Ahmad* ؛ Saira Hameed | ||
| Department of Mathematics, University of the Punjab, P.O.Box 54590, Lahore, Pakistan | ||
| چکیده | ||
| Let $\mathtt{A}(\mathtt{G})$ be the adjacency matrix of a simple connected undirected graph $\mathtt{G}$. A graph $\mathtt{G}$ of order $n$ is said to be non-singular (respectively singular) if $\mathtt{A}(\mathtt{G})$ is non-singular (respectively singular). The spectrum of a graph $\mathtt{G}$ is the set of all its eigenvalues denoted by $spec(\mathtt{G})$. The anti-reciprocal (respectively reciprocal) eigenvalue property for a graph $\mathtt{G}$ can be defined as `` Let $\mathtt{G}$ be a non-singular graph $\mathtt{G}$ if the negative reciprocal (respectively positive reciprocal) of each eigenvalue is likewise an eigenvalue of $\mathtt{G}$, then $\mathtt{G}$ has anti-reciprocal (respectively reciprocal) eigenvalue property ." Furthermore, a graph $\mathtt{G}$ is said to have strong anti-reciprocal eigenvalue property (resp. strong reciprocal eigenvalue property) if the eigenvalues and their negative (resp. positive) reciprocals are of same multiplicities. In this article, graphs satisfying anti-reciprocal eigenvalue (or property $(-\mathtt{R})$) and strong anti-reciprocal eigenvalue property (or property $(-\mathtt{SR})$) are discussed. | ||
| کلیدواژهها | ||
| Anti-reciprocal eigenvalue property؛ strong anti-reciprocal eigenvalue property؛ adjacency matrix؛ graph spectrum | ||
| مراجع | ||
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