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We consider the primitive two-colored digraphs whose uncolored digraph has $n+s$ vertices and consists of one $n$-cycle and one $(n-3)$-cycle. We give bounds on the exponents and characterizations of extremal two-colored digraphs.

The inertia set of a symmetric sign pattern $A$ is the set $i\left(A\right)=\{i\left(B\right)\mid B=B^T\in Q\left(A\right)\}$, where $i\left(B\right)$ denotes the inertia of real symmetric matrix $B$, and $Q\left(A\right)$ denotes the sign pattern class of $A$. In this paper, a complete characterization on the inertia set of the nonnegative symmetric sign pattern $A$ in which each diagonal entry is zero and all off-diagonal entries are positive is obtained. Further, we also consider the bound for the numbers of nonzero entries in the nonnegative symmetric sign patterns $A$ with zero diagonal that require...

The scrambling index of an $n\times n$ primitive Boolean matrix $A$ is the smallest positive integer $k$ such that $A^k{\left(A^{\mathrm{T}}\right)}^k=J$, where $A^{\mathrm{T}}$ denotes the transpose of $A$ and $J$ denotes the $n\times n$ all ones matrix. For an $m\times n$ Boolean matrix $M$, its Boolean rank $b\left(M\right)$ is the smallest positive integer $b$ such that $M=AB$ for some $m\times b$ Boolean matrix $A$ and $b\times n$ Boolean matrix $B$. In 2009, M. Akelbek, S. Fital, and J. Shen gave an upper bound on the scrambling index of an $n\times n$ primitive matrix $M$ in terms of its Boolean rank $b\left(M\right)$, and they also characterized all primitive...

Let $G$ be a simple connected graph with vertex set $V\left(G\right)=\{v_1,v_2,\cdots ,v_n\}$ and edge set $E\left(G\right)$, and let $d_{v_i}$ be the degree of the vertex $v_i$. Let $D\left(G\right)$ be the distance matrix and let $T_r\left(G\right)$ be the diagonal matrix of the vertex transmissions of $G$. The generalized distance matrix of $G$ is defined as $D_{\alpha }\left(G\right)=\alpha T_r\left(G\right)+(1-\alpha )D\left(G\right)$, where $0\le \alpha \le 1$. Let ${\lambda }_1\left(D_{\alpha }\left(G\right)\right)\ge {\lambda }_2\left(D_{\alpha }\left(G\right)\right)\ge ...\ge {\lambda }_n\left(D_{\alpha }\left(G\right)\right)$ be the generalized distance eigenvalues of $G$, and let $k$ be an integer with $1\le k\le n$. We denote by $S_k\left(D_{\alpha }\left(G\right)\right)={\lambda }_1\left(D_{\alpha }\left(G\right)\right)+{\lambda }_2\left(D_{\alpha }\left(G\right)\right)+...+{\lambda }_k\left(D_{\alpha }\left(G\right)\right)$ the sum of the $k$ largest generalized distance eigenvalues. The generalized distance spread of a graph $G$ is defined as $D_{\alpha }S\left(G\right)={\lambda }_1\left(D_{\alpha }\left(G\right)\right)-{\lambda }_n\left(D_{\alpha }\left(G\right)\right)$. We obtain some...

A sign pattern $A$ is a $\pm$ sign pattern if $A$ has no zero entries. $A$ allows orthogonality if there exists a real orthogonal matrix $B$ whose sign pattern equals $A$. Some sufficient conditions are given for a sign pattern matrix to allow orthogonality, and a complete characterization is given for $\pm$ sign patterns with $n-1\le N_{-}\left(A\right)\le n+1$ to allow orthogonality.

An $n\times n$ sign pattern $𝒜$ is said to be potentially nilpotent if there exists a nilpotent real matrix $B$ with the same sign pattern as $𝒜$. Let ${𝒟}_{n,r}$ be an $n\times n$ sign pattern with $2\le r\le n$ such that the superdiagonal and the $(n,n)$ entries are positive, the $(i,1)$ $(i=1,\cdots ,r)$ and $(i,i-r+1)$ $(i=r+1,\cdots ,n)$ entries are negative, and zeros elsewhere. We prove that for $r\ge 3$ and $n\ge 4r-2$, the sign pattern ${𝒟}_{n,r}$ is not potentially nilpotent, and so not spectrally arbitrary.

An $n\times n$ ray pattern $𝒜$ is called a spectrally arbitrary ray pattern if the complex matrices in $Q\left(𝒜\right)$ give rise to all possible complex polynomials of degree $n$. In a paper of Mei, Gao, Shao, and Wang (2014) was proved that the minimum number of nonzeros in an $n\times n$ irreducible spectrally arbitrary ray pattern is $3n-1$. In this paper, we introduce a new family of spectrally arbitrary ray patterns of order $n$ with exactly $3n-1$ nonzeros.

A sign pattern matrix (or nonnegative sign pattern matrix) is a matrix whose entries are from the set $\{+,-,0\}$ ($\{+,0\}$, respectively). The minimum rank (or rational minimum rank) of a sign pattern matrix $𝒜$ is the minimum of the ranks of the matrices (rational matrices, respectively) whose entries have signs equal to the corresponding entries of $𝒜$. Using a correspondence between sign patterns with minimum rank $r\ge 2$ and point-hyperplane configurations in ${ℝ}^{r-1}$ and Steinitz’s theorem on the rational realizability of...