solving graph coloring problem using graph adjacency matrix algorithm

‎graph coloring is the assignment of one color to each vertex of a graph so that two adjacent vertices are not of the same color‎. ‎the graph coloring problem (gcp) is a matter of combinatorial optimization‎, ‎and the goal of gcp is determining the chromatic number $chi(g)$‎. ‎since gcp is an np-hard problem‎, ‎then in this paper‎, ‎we propose a new approximated algorithm for finding the coloring number (it is an approximation of chromatic number) by using a graph adjacency matrix to colorize or separate a graph‎. ‎to prove the correctness of the proposed algorithm‎, ‎we implement it in matlab software‎, ‎and for analysis in terms of solution and execution time‎, ‎we compare our algorithm with some of the best existing algorithms that are already implemented in matlab software‎, ‎and we present the results in tables of various graphs‎. ‎several available algorithms used the largest degree selection strategy‎, ‎while our proposed algorithm uses the graph adjacency matrix to select the vertex that has the smallest degree for coloring‎. ‎we provide some examples to compare the performance of our algorithm to other available methods‎. ‎we make use of the dolan-mor’e performance profiles to assess the performance of the numerical algorithms‎, ‎and demonstrate the efficiency of our proposed approach in comparison with some existing methods‎.