Open AccessBounds on the $A_{\alpha}$-spread of a graph
Author(s) -
Zhen Lin,
Lianying Miao,
Shu-Guang Guo
Publication year - 2020
Publication title -
the electronic journal of linear algebra
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.447
H-Index - 31
eISSN - 1537-9582
pISSN - 1081-3810
DOI - 10.13001/ela.2020.5137
Subject(s) - adjacency matrix , mathematics , combinatorics , eigenvalues and eigenvectors , alpha (finance) , diagonal , diagonal matrix , graph , graph energy , matrix (chemical analysis) , discrete mathematics , line graph , graph power , physics , statistics , construct validity , geometry , materials science , quantum mechanics , composite material , psychometrics
Let $G$ be a simple undirected graph. For any real number $\alpha \in[0,1]$, Nikiforov defined the $A_{\alpha}$-matrix of $G$ as $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $A(G)$ and $D(G)$ are the adjacency matrix and the degree diagonal matrix of $G$, respectively. The $A_{\alpha}$-spread of a graph is defined as the difference between the largest eigenvalue and the smallest eigenvalue of the associated $A_{\alpha}$-matrix. In this paper, some lower and upper bounds on $A_{\alpha}$-spread are obtained, which extend the results of $A$-spread and $Q$-spread. Moreover, the trees with the minimum and the maximum $A_{\alpha}$-spread are determined, respectively.