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تسجيل مستخدم جديدThe inertia set of a signed graph
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A signed graph is a pair $(G,Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges are permitted, but loops are not) with $V={1,...,n}$ and $Sigmasubseteq E$. By $S(G,Sigma)$ we denote the set of all symmetric $Vtimes V$ matrices $A=[a_{i,j}]$ with $a_{i,j}<0$ if $i$ and $j$ are connected by only even edges, $a_{i,j}>0$ if $i$ and $j$ are connected by only odd edges, $a_{i,j}in mathbb{R}$ if $i$ and $j$ are connected by both even and odd edges, $a_{i,j}=0$ if $i ot=j$ and $i$ and $j$ are non-adjacent, and $a_{i,i} in mathbb{R}$ for all vertices $i$. The stable inertia set of a signed graph $(G,Sigma)$ is the set of all pairs $(p,q)$ for which there exists a matrix $Ain S(G,Sigma)$ with $p$ positive and $q$ negative eigenvalues which has the Strong Arnold Property. In this paper, we study the stable inertia set of (signed) graphs.
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A signed graph is a pair $(G,Sigma)$, where $G=(V,E)$ is a graph (in which parallel edges and loops are permitted) with $V={1,ldots,n}$ and $Sigmasubseteq E$. The edges in $Sigma$ are called odd edges and the other edges of $E$ even. By $S(G,Sigma)$
we denote the set of all symmetric $ntimes n$ real matrices $A=[a_{i,j}]$ such that if $a_{i,j} < 0$, then there must be an even edge connecting $i$ and $j$; if $a_{i,j} > 0$, then there must be an odd edge connecting $i$ and $j$; and if $a_{i,j} = 0$, then either there must be an odd edge and an even edge connecting $i$ and $j$, or there are no edges connecting $i$ and $j$. (Here we allow $i=j$.) For a symmetric real matrix $A$, the partial inertia of $A$ is the pair $(p,q)$, where $p$ and $q$ are the number of positive and negative eigenvalues of $A$, respectively. If $(G,Sigma)$ is a signed graph, we define the emph{inertia set} of $(G,Sigma)$ as the set of the partial inertias of all matrices $A in S(G,Sigma)$. In this paper, we present a formula that allows us to obtain the minimal elements of the inertia set of $(G,Sigma)$ in case $(G,Sigma)$ has a $1$-separation using the inertia sets of certain signed graphs associated to the $1$-separation.
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