It is known exactly what periods are possible on certain classes of graphs, such as trees Bitar and Goles (1992), simple cycles Dall'Asta (2006), the complete graph Levine (2008), and the

complete bipartite graph Jiang (2010).

i] of G by a complete bipartite graph [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for some [m.

Vaidya [7] proved that arbitrary supersubdivision of any tree, grid graph, complete bipartite graph star and [C.

A

complete bipartite graph with [absolute value of U]= [a-z] and [absolute value of V]= [a-z] is denoted by [K.

m,n,] m [less than or equal to] n be a

complete bipartite graph with m + n = p.

Up to now, the spectra are known for some particular graphs: path, cycle, complete graph,

complete bipartite graph, complete tree, hypercube, k-dimensional lattice, star graph, etc.

So we obtain that G is the

complete bipartite graph [K.

Finally, in Section 5 we give a formula for t(G; 2, -1) when G is a

complete bipartite graph, [K.

We are going to be looking at a subgraph of the

complete bipartite graph [K.

As pointed out in Haglund (2000) the MHH conjecture can be viewed as the analog of the MCP theorem from the

complete bipartite graph, [K.

We can see that the graph is composed of two

complete bipartite graphs, i.

Integrality and Laplacian integrality typically arise from tightly controlled combinatorial structure in special families of graphs, including complete graphs,

complete bipartite graphs and hypercubes (classical; see, e.