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A uniform attachment graph (with parameter $k$), denoted $G_{n,k}$ in thepaper, is a random graph on the vertex set $[n]$, where each vertex $v$ makes$k$ selections from $[v-1]$ uniformly and independently, and these selectionsdetermine the edge set. We study several aspects of this graph. Our motivationcomes from two similarly constructed, well-studied random graphs: $k$-outgraphs and preferential attachment graphs. In this paper, we find theasymptotic distribution of its minimum degree and connectivity, and study theexpansion properties of $G_{n,k}$ to show that the conductance of $G_{n,k}$ isof order $(\log n)^{-1}$. We also study the bootstrap percolation on $G_{n,k}$,where, each vertex is either initially infected with probability $p$,independently of others, or gets infected later as a result of having $r$infected neighbors at some point. We show that, for $2\le r\le k-1$, if $p\ll(\log n)^{-r/(r-1)}$, then, with probability approaching 1, the process endsbefore all vertices get infected. On the other hand, if $p\ge \omega(\logn)^{-r/(r-1)}$, where $\omega$ is a certain very slowly growing function, thenall the vertices get infected with probability approaching 1.

On connectivity, conductance and bootstrap percolation for a random k-out, age-biased graph