Applications of Estrada indices and energy to a family of compound graphs

作者：Andrade Enide^*; Pizarro Pamela; Robbiano Maria; San Martin B; Tapia Katherine

来源：Linear Algebra and Its Applications, 2017, 532: 162-178.

DOI：10.1016/j.laa.2017.06.035

• alpha-Adjacency matrix
• Universal matrix
• Compound graph
• Estrada index
• Signless Laplacian Estrada index
• Laplacian Estrada index
• Hypoenergetic graph
• Isospectral graph

To track the gradual change of the adjacency matrix of a simple graph g into the signless Laplacian matrix, V. Nikiforov in [35] suggested the study of the convex linear combination A(alpha) (alpha-adjacency matrix),
A(alpha) (g) = alpha D (g) + (1 - alpha) A (g),
for alpha is an element of [0,1], where A (g) and D (g) are the adjacency and the diagonal vertex degrees matrices of g, respectively. Taking this definition as an idea the next matrix was considered for a, b is an element of R. The matrix A(a,b) defined by
A(a,b) (g) = aD (g) + bA (g),
extends the previous alpha-adjacency matrix. This matrix is designated the (a, b)-adjacency matrix of g. Both adjacency matrices are examples of universal matrices already studied by W. Haemers. In this paper, we study the (a, b)-adjacency spectra for a family of compound graphs formed by disjoint balanced trees whose roots are identified to the vertices of a given graph. In consequence, new families of cospectral (adjacency, Laplacian and signless Laplacian) graphs, new hypoenergetic graphs (graphs whose energy is less than its vertex number) and new explicit formulae for Estrada, signless Laplacian Estrada and Laplacian Estrada indices of graphs were obtained. Moreover, sharp upper bounds of the above indices for caterpillars, in terms of length of the path and of the maximum number of its pendant vertices, are given.

• 出版日期2017-11-1