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The $A_{\alpha}$- spectrum of graph product
- Source :
- The Electronic Journal of Linear Algebra. 35:473-481
- Publication Year :
- 2019
- Publisher :
- University of Wyoming Libraries, 2019.
-
Abstract
- Let $A(G)$ and $D(G)$ denote the adjacency matrix and the diagonal matrix of vertex degrees of $G$, respectively. Define $$ A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G) $$ for any real $\alpha\in [0,1]$. The collection of eigenvalues of $A_{\alpha}(G)$ together with multiplicities is called the $A_{\alpha}$-\emph{spectrum} of $G$. Let $G\square H$, $G[H]$, $G\times H$ and $G\oplus H$ be the Cartesian product, lexicographic product, directed product and strong product of graphs $G$ and $H$, respectively. In this paper, a complete characterization of the $A_{\alpha}$-spectrum of $G\square H$ for arbitrary graphs $G$ and $H$, and $G[H]$ for arbitrary graph $G$ and regular graph $H$ is given. Furthermore, $A_{\alpha}$-spectrum of the generalized lexicographic product $G[H_1,H_2,\ldots,H_n]$ for $n$-vertex graph $G$ and regular graphs $H_i$'s is considered. At last, the spectral radii of $A_{\alpha}(G\times H)$ and $A_{\alpha}(G\oplus H)$ for arbitrary graph $G$ and regular graph $H$ are given.
- Subjects :
- Vertex (graph theory)
Algebra and Number Theory
010103 numerical & computational mathematics
Cartesian product
01 natural sciences
Combinatorics
symbols.namesake
Strong product of graphs
Diagonal matrix
symbols
Regular graph
Adjacency matrix
0101 mathematics
Graph product
Eigenvalues and eigenvectors
Mathematics
Subjects
Details
- ISSN :
- 10813810
- Volume :
- 35
- Database :
- OpenAIRE
- Journal :
- The Electronic Journal of Linear Algebra
- Accession number :
- edsair.doi...........976b3470b0a8d40aadfe3aad366371c2