arXiv subjects
Publications and source records attributed to Asghar Bahmani.
Let $T$ be a tree with a given adjacency eigenvalue $λ$. In this paper, by using the $λ$-minimal trees, we determine the structure of trees with a given multiplicity of the eigenvalue $λ$. Furthermore, we consider the relationship between the structure of trees and the eigensystem of a given Laplacian eigenvalue.
We give a counterexample to a conjecture of Wang and Hou related with the sum of the $k$ largest Laplacian eigenvalues of signed graphs.
Let $G$ be a simple graph with the Laplacian matrix $L(G)$ and let $e(G)$ be the number of edges of $G$. A conjecture by Brouwer and a conjecture by Grone and Merris state that the sum of the $k$ largest Laplacian eigenvalues of $G$ is at most $e(G)+\binom{k+1}{2}$ and $\sum_{i=1}^{k}d_{i}^{*}$, respectively, where $(d_{i}^{*})_{i}$ is the conjugate of the degree sequence $(d_i)_{i}$. We generalize these conjectures to weighted graphs and symmetric matrices. Moreover, among other results we show that under some assumptions, concave upper bounds on convex functions of symmetric real matrices are equivalent to concave upper bounds on convex functions of $(0,1)$ matrices.
Let $A$ be an acyclic symmetric matrix of order $n$. There is a weighted forest $F$ whose adjacency matrix is $A$. In this paper, using some results on matching polynomials, we provide an explicit formula for eigenvectors of $A$.
Let $M=[m_{ij}]$ be an $n\times m$ real matrix, $ρ$ be a nonzero real number, and $A$ be a symmetric real matrix. We denote by $D(M)$ the $n\times n$ diagonal matrix $diag(\sum_{j=1}^{m}m_{1j},\ldots,\sum_{j=1}^{m}m_{nj})$ and denote by $L_{A}^ρ$ the generalized Laplacian matrix $D(A)-ρA$. A well-known result of Grone et al. states that by connecting one of the end-vertices of $P_{3}$ to an arbitrary vertex of a graph, does not change the multiplicity of Laplacian eigenvalue $1$. We extend this theorem and some other results for a given generalized Laplacian eigenvalue $μ$. Furthermore, we give two proofs for a conjecture by Saito and Woei on the relation between the multiplicity of some Laplacian eigenvalues and pendant paths.