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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...

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) is a matrix whose entries are from the set {+, −, 0} and a sign vector is a vector whose entries are from the set {+, −, 0}. A sign pattern or sign vector is full if it does not contain any zero entries. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. The notions of essential row sign change number and essential column sign change number are introduced for full sign...

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...