Constraint programming models and population-based simulated annealing algorithm for finding graceful and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-labeling of quadratic graphs

In this research, a new mathematical integer programming model is presented for the graph labeling problem of quadratic graphs. The advantages of this model are linearity and the existence of an objective function. Furthermore, two constraint programming models and a meta-heuristics algorithm are also developed to generate feasible graceful labeling and α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-labeling for special classes of quadratic graphs. Experimental results on large sizes of graphs from the literature show the efficiency of the proposed model and approach.