Let $G$ be a connected simple graph on $n$ vertices. The Laplacian index of $G$, namely, the greatest Laplacian eigenvalue of $G$, is well known to be bounded above by $n$. In this paper, we give structural characterizations for graphs $G$ with the largest Laplacian index $n$. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on $n$ and $k$ for the existence of a $k$-regular graph $G$ of order $n$ with the largest Laplacian...

A well-known theorem of Rabin yields a dimensional lower bound on the width of complete polynomial proofs of a system of linear algebraic inequalities. In this note we investigate a practically motivated class of systems where the same lower bound can be obtained on the width of almost all (noncomplete) linear proofs. The proof of our result is based on the Helly Theorem.

In this paper the definition of Hermite-Hermite matrix polynomials is introduced starting from the Hermite matrix polynomials. An explicit representation, a matrix recurrence relation for the Hermite-Hermite matrix polynomials are given and differential equations satisfied by them is presented. A new expansion of the matrix exponential for a wide class of matrices in terms of Hermite-Hermite matrix polynomials is proposed.

A well-known theorem due to Kolchin states that a semi-group G of unipotent matrices over a field F can be brought to a triangular form over the field F [4, Theorem H]. Recall that a matrix A is called unipotent if its only eigenvalue is 1, or, equivalently, if the matrix I - A is nilpotent.Many years ago I noticed that this result of Kolchin is an immediate consequence of a too-little known result due to Wedderburn [6]. This result of Wedderburn asserts that if B is a finite dimensional algebra...

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 ${𝔹}_k$ be the general Boolean algebra and $T$ a linear operator on $M_{m,n}\left({𝔹}_k\right)$. If for any $A$ in $M_{m,n}\left({𝔹}_k\right)$ ($M_n\left({𝔹}_k\right)$, respectively), $A$ is regular (invertible, respectively) if and only if $T\left(A\right)$ is regular (invertible, respectively), then $T$ is said to strongly preserve regular (invertible, respectively) matrices. In this paper, we will give complete characterizations of the linear operators that strongly preserve regular (invertible, respectively) matrices over ${𝔹}_k$. Meanwhile, noting that a general Boolean algebra ${𝔹}_k$ is isomorphic...

For $X,Y\in {𝐌}_{n,m}$ it is said that $X$ is gut-majorized by $Y$, and we write $X{\prec }_{\mathrm{gut}}Y$, if there exists an $n$-by-$n$ upper triangular g-row stochastic matrix $R$ such that $X=RY$. Define the relation ${\sim }_{\mathrm{gut}}$ as follows. $X{\sim }_{\mathrm{gut}}Y$ if $X$ is gut-majorized by $Y$ and $Y$ is gut-majorized by $X$. The (strong) linear preservers of ${\prec }_{\mathrm{gut}}$ on ${ℝ}^n$ and strong linear preservers of this relation on ${𝐌}_{n,m}$ have been characterized before. This paper characterizes all (strong) linear preservers and strong linear preservers of ${\sim }_{\mathrm{gut}}$ on ${ℝ}^n$ and ${𝐌}_{n,m}$.

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.