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A sign pattern is an array with entries in $$\{+,-,0\}$$. A real matrix $$Q$$ is row orthogonal if $QQ^T = I$. The Strong Inner Product Property (SIPP), introduced in [B.A. Curtis and B.L. Shader, Sign patterns of orthogonal matrices and the strong inner product property, Linear Algebra Appl. 592: 228-259, 2020], is an important tool when determining whether a sign pattern allows row orthogonality because it guarantees there is a nearby matrix with the same property, allowing zero entries to be perturbed to nonzero entries, while preserving the sign of every nonzero entry. This paper uses the SIPP to initiate the study of conditions under which random sign patterns allow row orthogonality with high probability. Building on prior work, $$5\times n$$ nowhere zero sign patterns that minimally allow orthogonality are determined. Conditions on zero entries in a sign pattern are established that guarantee any row orthogonal matrix with such a sign pattern has the SIPP. 

A unified approach to the determination of eigenvalues and eigenvectors of specific matrices associated with directed graphs is presented. Matrices studied include the new distance matrix, with natural extensions to the distance Laplacian and distance signless Laplacian, in addition to the new adjacency matrix, with natural extensions to the Laplacian and signless Laplacian. Various sums of Kronecker products of nonnegative matrices are introduced to model the Cartesian and lexicographic products of digraphs. The Jordan canonical form is applied extensively to the analysis of spectra and eigenvectors. The analysis shows that Cartesian products provide a method for building infinite families of transmission regular digraphs with few distinct distance eigenvalues. 

The distance matrix $$\mathcal{D}(G)$$ of a graph $$G$$ is the matrix containing the pairwise distances between vertices, and the distance Laplacian matrix is $$\mathcal{D}^L (G)=T(G)-\mathcal{D} (G)$$, where $T(G)$ is the diagonal matrix of row sums of $$\mathcal{D}(G)$$. Several general methods are established for producing $$\mathcal{D}^L$$-cospectral graphs that can be used to construct infinite families. Examples are provided to show that various properties are not preserved by $$\mathcal{D}^L$$-cospectrality, including examples of $$\mathcal{D}^L$$-cospectral strongly regular and circulant graphs. It is established that the absolute values of coefficients of the distance Laplacian characteristic polynomial are decreasing, i.e., $$|\delta^L_{1}|\geq \cdots \geq |\delta^L_{n}|$$, where $$\delta^L_{k}$$ is the coefficient of $x^k$.