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A sign pattern requires a unique inertia if every real matrix in the sign pattern class has the same inertia. Several sufficient or necessary conditions are given for a sign pattern to require a unique inertia. It is proved that a sign pattern requires a unique inertia if and only if it requires a unique refined inertia. All sign patterns of orders 2 and 3 that require a unique inertia are characterized. If the underlying graph of a sign pattern is a tree, then it is shown that any skew-symmetric sign pattern requires a unique inertia, and a linear-time algorithm is given to determine whether or not a symmetric sign pattern requires a unique inertia. [ABSTRACT FROM AUTHOR]

The maximum nullity M ( G ) and the Colin de Verdière type parameter ξ ( G ) both consider the largest possible nullity over matrices in S ( G ) , which is the family of real symmetric matrices whose i , j -entry, i ≠ j , is nonzero if i is adjacent to j , and zero otherwise; however, ξ ( G ) restricts to those matrices A in S ( G ) with the Strong Arnold Property, which means X = O is the only symmetric matrix that satisfies A ∘ X = O , I ∘ X = O , and A X = O . This paper introduces zero forcing parameters Z SAP ( G ) and Z vc ( G ) , and proves that Z SAP ( G ) = 0 implies every matrix A ∈ S ( G ) has the Strong Arnold Property and that the inequality M ( G ) − Z vc ( G ) ≤ ξ ( G ) holds for every graph G . Finally, the values of ξ ( G ) are computed for all graphs up to 7 vertices, establishing ξ ( G ) = ⌊ Z ⌋ ( G ) for these graphs. [ABSTRACT FROM AUTHOR]

The distance matrix of a graph G is the matrix containing the pairwise distances between vertices. The distance eigenvalues of G are the eigenvalues of its distance matrix and they form the distance spectrum of G . We determine the distance spectra of double odd graphs and Doob graphs, completing the determination of distance spectra of distance regular graphs having exactly one positive distance eigenvalue. We characterize strongly regular graphs having more positive than negative distance eigenvalues. We give examples of graphs with few distinct distance eigenvalues but lacking regularity properties. We also determine the determinant and inertia of the distance matrices of lollipop and barbell graphs. [ABSTRACT FROM AUTHOR]