This paper gives some new results on mutually orthogonal graph squares (MOGS). These generalize mutually orthogonal Latin squares in an interesting way. As such, the topic is quite nice and should have broad appeal. MOGS have strong connections to core fields of finite algebra, cryptography, finite geometry, and design of experiments. We are concerned with the Kronecker product of mutually orthogonal graph squares to get new results of the mutually orthogonal certain graphs squares.

For Cauchy numbers of the first kind \(\{a_n\}_{n \geq 0}\) and Cauchy numbers of the second kind \(\{b_n\}_{n \geq 0}\), we prove that two sequences \(\left\{ \sqrt[n]{|a_n|} \right\}_{n \geq 2}\) and \(\left\{ \sqrt[n]{b_n} \right\}_{n \geq 1}\) are log-concave. In addition, we show that two sequences \(\left\{ \frac{1}{\sqrt[n]{|a_n|}} \right\}_{n \geq 2}\) and \(\left\{ \frac{1}{\sqrt[n]{b_n}} \right\}_{n \geq 1}\) are log-balanced.

Let \( p(x) = a_0 + a_1x + \dots + a_nx^n \) be a polynomial with all roots real and satisfying \( x \leq -\delta \) for some \( 0 < \delta < 1 \). We show that for any \( 0 < \epsilon 0 \). As a corollary, we show that if \( m_k(G) \) is the number of matchings with \( k \) edges in a graph \( G \), then for any \( 0 < \epsilon 0 \) is an absolute constant. We prove a similar result for polynomials with complex roots satisfying \( \Re z \leq -\delta \) and apply it to estimate the number of unbranched subgraphs of \( G \).

Let \( G \) be a graph, a subset \( S \subseteq E(G) \) is called an edge hub set of \( G \) if every pair of edges \( e, f \in E(G) \setminus S \) are connected by a path where all internal edges are from \( S \). The minimum cardinality of an edge hub set is called the edge hub number of \( G \), and is denoted by \( h_e(G) \). If \( G \) is a disconnected graph, then any edge hub set must contain all of the edges in all but one of the components, as well as an edge hub set in the remaining component. In this paper, the edge hub number for several classes of graphs is computed, and bounds in terms of other graph parameters are also determined.

In 1998, D. Callan obtained a binomial identity involving the derangement numbers. In this paper, by using the theory of formal series, we extend such an identity to the generalized derangement numbers. Then, by using the same technique, we obtain other identities of the same kind for the generalized arrangement numbers, the generalized Laguerre polynomials, the generalized Hermite polynomials, the generalized exponential polynomials and the generalized Bell numbers, the hyperharmonic numbers, the Lagrange polynomials and the Gegenbauer polynomials.