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Let $mathbb{F}$ be an infinite field with characteristic different from two. For a graph $G=(V,E)$ with $V={1,...,n}$, let $S(G;mathbb{F})$ be the set of all symmetric $ntimes n$ matrices $A=[a_{i,j}]$ over $mathbb{F}$ with $a_{i,j} ot=0$, $i ot=j$ if and only if $ijin E$. We show that if $G$ is the complement of a partial $k$-tree and $mgeq k+2$, then for all nonsingular symmetric $mtimes m$ matrices $K$ over $mathbb{F}$, there exists an $mtimes n$ matrix $U$ such that $U^T K Uin S(G;mathbb{F})$. As a corollary we obtain that, if $k+2leq mleq n$ and $G$ is the complement of a partial $k$-tree, then for any two nonnegative integers $p$ and $q$ with $p+q=m$, there exists a matrix in $S(G;reals)$ with $p$ positive and $q$ negative eigenvalues.
Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which in
The inverse eigenvalue problem of a graph $G$ aims to find all possible spectra for matrices whose $(i,j)$-entry, for $i eq j$, is nonzero precisely when $i$ is adjacent to $j$. In this work, the inverse eigenvalue problem is completely solved for a
In this paper, we use a new and correct method to determine the $n$-vertex $k$-trees with the first three largest signless Laplacian indices.
In this paper, we present an involution on some kind of colored $k$-ary trees which provides a combinatorial proof of a combinatorial sum involving the generalized Catalan numbers $C_{k,gamma}(n)=frac{gamma}{k n+gamma}{k n+gammachoose n}$. From the c
The inverse eigenvalue problem of a given graph $G$ is to determine all possible spectra of real symmetric matrices whose off-diagonal entries are governed by the adjacencies in $G$. Barrett et al. introduced the Strong Spectral Property (SSP) and th