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In this short note we provide an extension of the notion of Hessenberg matrix and observe an identity between the determinant and the permanent of such matrices. The celebrated identity due to Gibson involving Hessenberg matrices is consequently generalized.

Suppose that $A$ is a real symmetric matrix of order $n$. Denote by $m_A\left(0\right)$ the nullity of $A$. For a nonempty subset $\alpha$ of $\{1,2,...,n\}$, let $A\left(\alpha \right)$ be the principal submatrix of $A$ obtained from $A$ by deleting the rows and columns indexed by $\alpha$. When $m_{A\left(\alpha \right)}\left(0\right)=m_A\left(0\right)+\left|\alpha \right|$, we call $\alpha$ a P-set of $A$. It is known that every P-set of $A$ contains at most $\lfloor n/2\rfloor$ elements. The graphs of even order for which one can find a matrix attaining this bound are now completely characterized. However, the odd case turned out to be more difficult to tackle. As a first step...

We analyse the roots of the polynomial $x^n-p{x}^{n-1}-qx-1$ for $p⩾q⩾1$. This is the characteristic polynomial of the recurrence relation $F_{k,p,q}\left(n\right)=p{F}_{k,p,q}(n-1)+q{F}_{k,p,q}(n-k+1)+F_{k,p,q}(n-k)$ for $n⩾k$, which includes the relations of several particular sequences recently defined. In the end, a matricial representation for such a recurrence relation is provided.