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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 inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, and investigate restrictions on the inertia set of any graph.
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 $a_{i,j} ot=0$, $i ot=j$ i
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
For a graph $G$, we associate a family of real symmetric matrices, $mathcal{S}(G)$, where for any $M in mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ are governed by the adjacency structure of $G$. The ordered multiplicity I
A caterpillar graph $T(p_1, ldots, p_r)$ of order $n= r+sum_{i=1}^r p_i$, $rgeq 2$, is a tree such that removing all its pendent vertices gives rise to a path of order $r$. In this paper we establish a necessary and sufficient condition for a real nu
Ramanujan graphs have fascinating properties and history. In this paper we explore a parallel notion of Ramanujan digraphs, collecting relevant results from old and recent papers, and proving some new ones. Almost-normal Ramanujan digraphs are shown