Let $G=(V,E)$, $V=\{v_1,v_2,...,v_n\}$, be a simple connected graph with $n$ vertices, $m$ edges and a sequence of vertex degrees $d_1\ge d_2\ge \cdots \ge d_n$. Denote by $A$ and $D$ the adjacency matrix and diagonal vertex degree matrix of $G$, respectively. The signless Laplacian of $G$ is defined as $L^{+}=D+A$ and the normalized signless Laplacian matrix as ${ℒ}^{+}=D^{-1/2}{L}^{+}{D}^{-1/2}$. The normalized signless Laplacian spreads of a connected nonbipartite graph $G$ are defined as $r\left(G\right)={\gamma }_2^{+}/{\gamma }_n^{+}$ and $l\left(G\right)={\gamma }_2^{+}-{\gamma }_n^{+}$, where ${\gamma }_1^{+}\ge {\gamma }_2^{+}\ge \cdots \ge {\gamma }_n^{+}\ge 0$ are eigenvalues of ${ℒ}^{+}$. We establish sharp lower and upper bounds for the normalized signless Laplacian spreads...

A graph is determined by its signless Laplacian spectrum if no other non-isomorphic graph has the same signless Laplacian spectrum (simply $G$ is $DQS$). Let $T(a,b,c)$ denote the $T$-shape tree obtained by identifying the end vertices of three paths $P_{a+2}$, $P_{b+2}$ and $P_{c+2}$. We prove that its all line graphs $ℒ\left(T\right(a,b,c\left)\right)$ except $ℒ\left(T\right(t,t,2t+1\left)\right)$ ($t\ge 1$) are $DQS$, and determine the graphs which have the same signless Laplacian spectrum as $ℒ\left(T\right(t,t,2t+1\left)\right)$. Let ${\mu }_1\left(G\right)$ be the maximum signless Laplacian eigenvalue of the graph $G$. We give the limit of ${\mu }_1\left(ℒ\left(T(a,b,c)\right)\right)$, too.

We discuss the eigenvalue problem in the max-plus algebra. For a max-plus square matrix, the roots of its characteristic polynomial are not its eigenvalues. In this paper, we give the notion of algebraic eigenvectors associated with the roots of characteristic polynomials. Algebraic eigenvectors are the analogues of the usual eigenvectors in the following three senses: (1) An algebraic eigenvector satisfies an equation similar to the equation $A\otimes x=\lambda \otimes x$ for usual eigenvectors. Under a suitable assumption,...

A matrix $A$ in $(max,min)$-algebra (fuzzy matrix) is called weakly robust if $A^k\otimes x$ is an eigenvector of $A$ only if $x$ is an eigenvector of $A$. The weak robustness of fuzzy matrices are studied and its properties are proved. A characterization of the weak robustness of fuzzy matrices is presented and an $O\left(n^2\right)$ algorithm for checking the weak robustness is described.

A connection between representation of compact groups and some invariant ensembles of hermitian matrices is described. We focus on two types of invariant ensembles which extend the gaussian and the Laguerre Unitary ensembles. We study them using projections and convolutions of invariant probability measures on adjoint orbits of a compact Lie group. These measures are described by semiclassical approximation involving tensor and restriction multiplicities. We show that a large class of them are determinantal....

The spectral radius of a graph is defined by that of its unoriented Laplacian matrix. In this paper, we determine the unicyclic graphs respectively with the third and the fourth largest spectral radius among all unicyclic graphs of given order.