Remarks on Q -integral complete multipartite graphs

Integral complete multipartite graphs

A graph is called integral if all eigenvalues of its adjacency matrix are integers. In this paper, we investigate integral complete r-partite graphsKp1,p2,...,pr =Ka1·p1,a2·p2,...,as ·ps with s=3, 4.We can construct infinite many new classes of such integral graphs by solving some certain Diophantine equations. These results are different from those in the existing literature. For s = 4, we giv...

On graphs with complete multipartite µ-graphs

Complete subgraphs in multipartite graphs

Turán’s Theorem states that every graphG of edge density ‖G‖/ (|G| 2 ) > k−2 k−1 contains a complete graph K and describes the unique extremal graphs. We give a similar Theorem for `-partite graphs. For large `, we find the minimal edge density d` , such that every `-partite graph whose parts have pairwise edge density greater than d` contains a K . It turns out that d` = k−2 k−1 for large enou...

Decompositions of complete multipartite graphs

This paper answers a recent question of Dobson and Marušič by partitioning the edge set of a complete bipartite graph into two parts, both of which are edge sets of arctransitive graphs, one primitive and the other imprimitive. The first member of the infinite family is the one constructed by Dobson and Marušič.

Domination games played on line graphs of complete multipartite graphs

The domination game on a graph G (introduced by B. Brešar, S. Klavžar, D.F. Rall [1]) consists of two players, Dominator and Staller, who take turns choosing a vertex from G such that whenever a vertex is chosen by either player, at least one additional vertex is dominated. Dominator wishes to dominate the graph in as few steps as possible, and Staller wishes to delay this process as much as po...