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GRAPHS WITH CLUSTERS PERTURBED BY REGULAR GRAPHS--Aα-SPECTRUM AND APPLICATIONS.
- Source :
-
Discussiones Mathematicae: Graph Theory . 2020, Vol. 40 Issue 2, p451-466. 16p. - Publication Year :
- 2020
-
Abstract
- Given a graph G, its adjacency matrix A(G) and its diagonal matrix of vertex degrees D(G) consider the matrix Aα(G) = αD(G) + (1 -- α)A(G), where α ∊ [0,1). The Aα-spectrum of G is the multiset of eigenvalues of Aα(G) and these eigenvalues are the α-eigenvalues of G. A cluster in G is a pair of vertex subsets (C, S), where C is a set of cardinality |C| ≥ 2 of pairwise co-neighbor vertices sharing the same set S of |S| neighbors. Assuming that G is connected and it has a cluster (C, S), G(H) is obtained from G and an r-regular graph H of order |C| by identifying its vertices with the vertices in C, eigenvalues of Aα(G) and Aα(G(H)) are deduced and if Aα(H) is positive semidefinite, then the i-th eigenvalue of Aα(G(H)) is greater than or equal to i-th eigenvalue of Aα(G). These results are extended to graphs with several pairwise disjoint clusters (C1, S1), .. ., (Ck, Sk). As an application, the effect on the energy, α-Estrada index and α-index of a graph G with clusters when the edges of regular graphs are added to G are analyzed. Finally, the Aα-spectrum of the corona product G o H of a connected graph G and a regular graph H is determined. [ABSTRACT FROM AUTHOR]
- Subjects :
- *REGULAR graphs
*GRAPH connectivity
*EIGENVALUES
Subjects
Details
- Language :
- English
- ISSN :
- 12343099
- Volume :
- 40
- Issue :
- 2
- Database :
- Academic Search Index
- Journal :
- Discussiones Mathematicae: Graph Theory
- Publication Type :
- Academic Journal
- Accession number :
- 141502266
- Full Text :
- https://doi.org/10.7151/dmgt.2284