Geodesic
Graphs with Clusters Perturbed by Regular Graphs—Aα-Spectrum and Applications
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 451-466

Voir la notice de l'article provenant de la source Library of Science

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 ◦ H of a connected graph G and a regular graph H is determined.
Keywords: cluster, convex combination of matrices, corona product of graphs, Aα-spectrum 
@article{DMGT_2020_40_2_a6,
     author = {Cardoso, Domingos M. and Past\'en, Germain and Rojo, Oscar},
     title = {Graphs with {Clusters} {Perturbed} by {Regular} {Graphs{\textemdash}A\protect\textsubscript{\ensuremath{\alpha}}-Spectrum} and {Applications}},
     journal = {Discussiones Mathematicae. Graph Theory},
     pages = {451--466},
     publisher = {mathdoc},
     volume = {40},
     number = {2},
     year = {2020},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a6/}
}
TY  - JOUR
AU  - Cardoso, Domingos M.
AU  - Pastén, Germain
AU  - Rojo, Oscar
TI  - Graphs with Clusters Perturbed by Regular Graphs—Aα-Spectrum and Applications
JO  - Discussiones Mathematicae. Graph Theory
PY  - 2020
SP  - 451
EP  - 466
VL  - 40
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a6/
LA  - en
ID  - DMGT_2020_40_2_a6
ER  - 
%0 Journal Article
%A Cardoso, Domingos M.
%A Pastén, Germain
%A Rojo, Oscar
%T Graphs with Clusters Perturbed by Regular Graphs—Aα-Spectrum and Applications
%J Discussiones Mathematicae. Graph Theory
%D 2020
%P 451-466
%V 40
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a6/
%G en
%F DMGT_2020_40_2_a6
Cardoso, Domingos M.; Pastén, Germain; Rojo, Oscar. Graphs with Clusters Perturbed by Regular Graphs—Aα-Spectrum and Applications. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 2, pp. 451-466. http://geodesic.mathdoc.fr/item/DMGT_2020_40_2_a6/