We consider a k-tridiagonal ℓ-Toeplitz matrix as one of generalizations of a tridiagonal Toeplitz matrix. In the present paper, we provide a decomposition of the matrix under a certain condition. By the decomposition, the matrix is easily analyzed since one only needs to analyze the small matrix obtained from the decomposition. Using the decomposition, eigenpairs and arbitrary integer powers of the matrix are easily shown as applications.

For a simple connected graph $G$ of order $n$ having distance Laplacian eigenvalues ${\rho }_1^L\ge {\rho }_2^L\ge \cdots \ge {\rho }_n^L$, the distance Laplacian energy $\mathrm{DLE}\left(G\right)$ is defined as $\mathrm{DLE}\left(G\right)={\sum }_{i=1}^n|{\rho }_i^L-2W\left(G\right)/n|$, where $W\left(G\right)$ is the Wiener index of $G$. We obtain a relationship between the Laplacian energy and the distance Laplacian energy for graphs with diameter 2. We obtain lower bounds for the distance Laplacian energy $\mathrm{DLE}\left(G\right)$ in terms of the order $n$, the Wiener index $W\left(G\right)$, the independence number, the vertex connectivity number and other given parameters. We characterize the extremal graphs...

Let $G$ be a mixed graph. The eigenvalues and eigenvectors of $G$ are respectively defined to be those of its Laplacian matrix. If $G$ is a simple graph, [M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975), 619–633] gave a remarkable result on the structure of the eigenvectors of $G$ corresponding to its second smallest eigenvalue (also called the algebraic connectivity of $G$). For $G$ being a general mixed graph with...

Let $G$ be an undirected connected graph with $n$, $n\ge 3$, vertices and $m$ edges with Laplacian eigenvalues ${\mu }_1\ge {\mu }_2\ge \cdots \ge {\mu }_{n-1}>{\mu }_n=0$. Denote by ${\mu }_I={\mu }_{r_1}+{\mu }_{r_2}+\cdots +{\mu }_{r_k}$, $1\le k\le n-2$, $1\le r_1<r_2<\cdots <r_k\le n-1$, the sum of $k$ arbitrary Laplacian eigenvalues, with ${\mu }_{I_1}={\mu }_1+{\mu }_2+\cdots +{\mu }_k$ and ${\mu }_{I_n}={\mu }_{n-k}+\cdots +{\mu }_{n-1}$. Lower bounds of graph invariants ${\mu }_{I_1}-{\mu }_{I_n}$ and ${\mu }_{I_1}/{\mu }_{I_n}$ are obtained. Some known inequalities follow as a special case.

Let F be an analytic function from an open subset Ω of the complex plane into the algebra of n×n matrices. Denoting by $s_1,...,s_n$ the decreasing sequence of singular values of a matrix, we prove that the functions $log{s}_1\left(F\left(\lambda \right)\right)+...+log{s}_k\left(F\left(\lambda \right)\right)$ and $lo{g}^{+}{s}_1\left(F\left(\lambda \right)\right)+...+lo{g}^{+}{s}_k\left(F\left(\lambda \right)\right)$ are subharmonic on Ω for 1 ≤ k ≤ n.

Starting with Dürer's magic square which appears in the well-known copper plate engraving Melencolia we consider the class of melancholic magic squares. Each member of this class exhibits the same 86 patterns of Dürer's magic square and is magic again. Special attention is paid to the eigenstructure of melancholic magic squares, their group inverse and their Moore-Penrose inverse. It is seen how the patterns of the original Dürer square to a large extent are passed down also to the inverses of the...