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In this paper, we introduce a new generalization of Leonardo numbers, which are so-called Leonardo $p$-numbers. We investigate some basic properties of these numbers. We also define incomplete Leonardo $p$-numbers which generalize the incomplete Leonardo numbers.

A set of edges $X\subseteq E(G)$ of a graph $G$ is an edge general position set if no three edges from $X$ lie on a common shortest path. The edge general position number ${\rm gp}_{\rm e}(G)$ of $G$ is the cardinality of a largest edge general position set in $G$. Graphs $G$ with ${\rm gp}_{\rm e}(G) = |E(G)| - 1$ and with ${\rm gp}_{\rm e}(G) = 3$ are respectively characterized. Sharp upper and lower bounds on ${\rm gp}_{\rm e}(G)$ are proved for block graphs $G$ and exact values are determined for several specific block graphs.

In this paper, we give upper and lower bounds for the spectral norms of r-circulant matrices with the generalized bi-periodic Fibonacci numbers. Moreover, we investigate the eigenvalues and determinants of these matrices.

The hybrid numbers were introduced by Ozdemir [9] as a new generalization of complex, dual, and hyperbolic numbers. A hybrid number is defined by $k=a+bi+c\epsilon +dh$, where $a,b,c,d$ are real numbers and $% i,\epsilon ,h$ are operators such that $i^{2}=-1,\epsilon ^{2}=0,h^{2}=1$ and $ih=-hi=\epsilon +i$. This work is intended as an attempt to introduce the bi-periodic Horadam hybrid numbers which generalize the classical Horadam hybrid numbers. We give the generating function, the Binet formula, and some basic properties of these new hybrid numbers. Also, we investigate some relationships between generalized bi-periodic Fibonacci hybrid numbers and generalized bi-periodic\ Lucas hybrid numbers.