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We show that finding a graph realization with the minimum Randic index for a given degree sequence is solvable in polynomial time by formulating the problem as a minimum weight perfect b-matching problem. However, the realization found via this reduction is not guaranteed to be connected. Approximating the minimum weight b-matching problem subject to a connectivity constraint is shown to be NP-Hard. For instances in which the optimal solution to the minimum Randic index problem is not connected, we describe a heuristic to connect the graph using pairwise edge exchanges that preserves the degree sequence. In our computational experiments, the heuristic performs well and the Randic index of the realization after our heuristic is within 3% of the unconstrained optimal value on average. Although we focus on minimizing the Randic index, our results extend to maximizing the Randic index as well. Applications of the Randic index to synchronization of neuronal networks controlling respiration in mammals and to normalizing cortical thickness networks in diagnosing individuals with dementia are provided.
We give the first $2$-approximation algorithm for the cluster vertex deletion problem. This is tight, since approximating the problem within any constant factor smaller than $2$ is UGC-hard. Our algorithm combines the previous approaches, based on th
A split graph is a graph whose vertex set can be partitioned into a clique and a stable set. Given a graph $G$ and weight function $w: V(G) to mathbb{Q}_{geq 0}$, the Split Vertex Deletion (SVD) problem asks to find a minimum weight set of vertices $
A path in an(a) edge(vertex)-colored graph is called emph{a conflict-free path} if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called emph{conflict-free (vertex-)connected} if there is a conflict-
We study variants of Mastermind, a popular board game in which the objective is sequence reconstruction. In this two-player game, the so-called textit{codemaker} constructs a hidden sequence $H = (h_1, h_2, ldots, h_n)$ of colors selected from an alp
Traditionally, network analysis is based on local properties of vertices, like their degree or clustering, and their statistical behavior across the network in question. This paper develops an approach which is different in two respects. We investiga