arXiv subjects
Publications and source records attributed to Farkhondeh Sajadi.
In this paper we prove that the limiting distribution of the Chromatic number of a random graph $\mathcal{G}_{n,p}$, with fixed edge-probability $p$, after appropriate centering and scaling is Normal, when the number of vertices $n$, goes to infinity.
In this paper we consider a simple virus infection spread model on a finite population of $n$ agents connected by some neighborhood structure. Given a graph $G$ on $n$ vertices, we begin with some fixed number of initial infected vertices. At each discrete time step, an infected vertex tries to infect its neighbors with probability $β\in (0,1)$ independently of others and then it dies out. The process continues till all infected vertices die out. We focus on obtaining proper lower bounds on the expected number of ever infected vertices. We obtain a simple lower bound, using \textit{breadth-first search} algorithm and show that for a large class of graphs which can be classified as the ones which locally "look like" a tree in sense of the \emph{local weak convergence}, this lower bound gives better approximation than some of the known approximations through matrix-method based upper bounds derived by Draief, Ganesh and Massoulie in 2008.
In this work we consider the mean field traveling salesman problem, where the intercity distances are taken to be i.i.d. with some distribution $F$. This paper focus on the \emph{nearest neighbor tour} which is to move to the nearest non-visited city and we show that under some conditions on $F$, which are satisfied by exponential distribution with constant mean, the total length of the nearest neighbor tour, asymptotically almost surely scales as $\log n$. Similar result is known for Euclidean TSP and nearest neighbor tour. We further derive the limiting behavior of the total length of the nearest neighbor tour for more general distribution function $F$ and show that its asymptotic properties are determined by the scaling properties of the density of $F$ at 0.
For connectivity of \emph{random geometric graphs}, where there is no density for underlying distribution of the vertices, we consider $n$ i.i.d. \emph{Cantor} distributed points on $[0,1]$. We show that for this random geometric graph, the connectivity threshold $R_{n}$, converges almost surely to a constant $1-2ϕ$ where $0 < ϕ< 1/2$, which for the standard Cantor distribution is 1/3. We also show that $\| R_n - (1 - 2 ϕ) \|_1 \sim 2 \, C(ϕ) \, n^{-1/d_ϕ}$ where $C(ϕ) > 0$ is a constant and $d_ϕ := - {\log 2}/{\log ϕ}$ is the \emph{Hausdorff dimension} of the generalized Cantor set with parameter $ϕ$.