Let $A \in {\cal C}^n$ be an extremal copositive matrix with unit diagonal.
Then the minimal zeros of $A$ all have supports of cardinality two if and only
if the elements of $A$ are all from the set $\{-1,0,1\}$. Thus the extremal
copositive matrices with minimal zero supports of cardinality two are exactly
those matrices which can be obtained by diagonal scaling from the extremal
$\{-1,0,1\}$ unit diagonal matrices characterized by Hoffman and Pereira in
1973.Comment: 4 page

Hildebrand, Roland

English

arXiv

International audienceLet $A \in {\cal C}^n$ be an exceptional extremal copositive $n \times n$ matrix with positive diagonal. A zero $u$ of $A$ is a non-zero nonnegative vector such that $u^TAu = 0$. The support of a zero $u$ is the index set of the positive elements of $u$. A zero $u$ is minimal if there is no other zero $v$ such that ${\rm supp} v \subset {\rm supp} u$ strictly. Let $G$ be the graph on $n$ vertices which has an edge $(i,j)$ if and only if $A$ has a zero with support $\{1,\dots,n\} \setminus \{i,j\}$. In this paper, it is shown that $G$ cannot contain a cycle of length strictly smaller than $n$. As a consequence, if all minimal zeros of $A$ have support of cardinality $n - 2$, then $G$ must be the cycle graph $C_n$

Hal - Université Grenoble Alpes

Extremal Copositive Matrices with Zero Supports of Cardinality n-2

International audienceLet $A \in {\cal C}^n$ be an extremal copositive matrix with unit diagonal. Then the minimal zeros of $A$ all have supports of cardinality two if and only if the elements of $A$ are all from the set $\{-1,0,1\}$. Thus the extremal copositive matrices with minimal zero supports of cardinality two are exactly those matrices which can be obtained by diagonal scaling from the extremal $\{-1,0,1\}$ unit diagonal matrices characterized by Hoffman and Pereira in 1973

INRIA a CCSD electronic archive server

Extremal copositive matrices with minimal zero supports of cardinality two

Let $A \in {\cal C}^n$ be an exceptional extremal copositive $n \times n$ matrix with positive diagonal. A zero $u$ of $A$ is a non-zero nonnegative vector such that $u^TAu = 0$. The support of a zero $u$ is the index set of the positive elements of $u$. A zero $u$ is minimal if there is no other zero $v$ such that $\Supp v \subset \Supp u$ strictly. Let $G$ be the graph on $n$ vertices which has an edge $(i,j)$ if and only if $A$ has a zero with support $\{1,\dots,n\} \setminus \{i,j\}$. In this paper, it is shown that $G$ cannot contain a cycle of length strictly smaller than $n$. As a consequence, if all minimal zeros of $A$ have support of cardinality $n - 2$, then $G$ must be the cycle graph $C_n$

Extremal copositive matrices with minimal zero supports of cardinality two

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Hal - Université Grenoble Alpes

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Hal - Université Grenoble Alpes

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Hal - Université Grenoble Alpes

Archive Ouverte en Sciences de l'Information et de la Communication

INRIA a CCSD electronic archive server