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The problem of finding completely positive matrices with equal cp-rank and rank is considered. We give some easy-to-check sufficient conditions on the entries of a doubly nonnegative matrix for it to be completely positive with equal cp-rank and rank. An algorithm is also presented to show its efficiency.
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Let $N(leq m,n)$ denote the number of partitions of $n$ with rank not greater than $m$, and let $M(leq m,n)$ denote the number of partitions of $n$ with crank not greater than $m$. Bringmann and Mahlburg observed that $N(leq m,n)leq M(leq m,n)leq N(leq m+1,n)$ for $m<0$ and $1leq nleq 100$. They also pointed out that these inequalities can be restated as the existence of a re-ordering $tau_n$ on the set of partitions of $n$ such that $|text{crank}(lambda)|-|text{rank}(tau_n(lambda))|=0$ or $1$ for all partitions $lambda$ of $n$, that is, the rank and the crank are nearly equal distributions over partitions of $n$. In the study of the spt-function, Andrews, Dyson and Rhoades proposed a conjecture on the unimodality of the spt-crank, and they showed that this conjecture is equivalent to the inequality $N(leq m,n)leq M(leq m,n)$ for $m<0$ and $ngeq 1$. We proved this conjecture by combiantorial arguments. In this paper, we prove the inequality $N(leq m,n)leq M(leq m,n)$ for $m<0$ and $ngeq 1$. Furthermore, we define a re-ordering $tau_n$ of the partitions $lambda$ of $n$ and show that this re-ordering $tau_n$ leads to the nearly equal distribution of the rank and the crank. Using the re-ordering $tau_n$, we give a new combinatorial interpretation of the function ospt$(n)$ defined by Andrews, Chan and Kim, which immediately leads to an upper bound for $ospt(n)$ due to Chan and Mao.
We prove that strength and slice rank of homogeneous polynomials of degree $d geq 5$ over an algebraically closed field of characteristic zero coincide generically. To show this, we establish a conjecture of Catalisano, Geramita, Gimigliano, Harbourne, Migliore, Nagel and Shin concerning dimensions of secant varieties of the varieties of reducible homogeneous polynomials. These statements were already known in degrees $2leq dleq 7$ and $d=9$.
An $ntimes n$ matrix $M$ is called a textit{fooling-set matrix of size $n$} if its diagonal entries are nonzero and $M_{k,ell} M_{ell,k} = 0$ for every $k e ell$. Dietzfelbinger, Hromkovi{v{c}}, and Schnitger (1996) showed that $n le (mbox{rk} M)^2$, regardless of over which field the rank is computed, and asked whether the exponent on $mbox{rk} M$ can be improved. We settle this question. In characteristic zero, we construct an infinite family of rational fooling-set matrices with size $n = binom{mbox{rk} M+1}{2}$. In nonzero characteristic, we construct an infinite family of matrices with $n= (1+o(1))(mbox{rk} M)^2$.
Our main result is a sharp bound for the number of vertices in a minimal forbidden subgraph for the graphs having minimum rank at most 3 over the finite field of order 2. We also list all 62 such minimal forbidden subgraphs. We conclude by exploring how some of these results over the finite field of order 2 extend to arbitrary fields and demonstrate that at least one third of the 62 are minimal forbidden subgraphs over an arbitrary field for the class of graphs having minimum rank at most 3 in that field.
The rank of a graph is defined to be the rank of its adjacency matrix. A graph is called reduced if it has no isolated vertices and no two vertices with the same set of neighbors. A reduced graph $G$ is said to be maximal if any reduced graph containing $G$ as a proper induced subgraph has a higher rank. The main intent of this paper is to present some results on maximal graphs. First, we introduce a characterization of maximal trees (a reduced tree is a maximal tree if it is not a proper subtree of a reduced tree with the same rank). Next, we give a near-complete characterization of maximal `generalized friendship graphs. Finally, we present an enumeration of all maximal graphs with ranks $8$ and $9$. The ranks up to $7$ were already done by Lepovic (1990), Ellingham (1993), and Lazic (2010).
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