The main aim of this study is to characterize new classes of multicone graphs which are determined by their signless Laplacian spectra. A multicone graph is defined to be the join of a clique and a regular graph. Let C and K w denote the Clebsch graph and a complete graph on w vertices, respectively. In this paper, we show that the multicone graphs K w ▽C are determined by their signless ...

In the course of extending Grötzsch’s theorem, we prove that every triangle-free graph without a K5-minor is 3-colorable. It has been recently proved that every triangle-free planar graph admits a homomorphism to the Clebsch graph. We also extend this result to the class of triangle-free graphs without a K5-minor. This is related to some conjectures which generalize the Four-Color Theorem. Whil...

The graphs with no five-vertex induced path are still not understood. But in the triangle-free case, we can do this and one better; we give an explicit construction for all triangle-free graphs with no six-vertex induced path. Here are two examples: the 16-vertex Clebsch graph, and the graph obtained from a complete bipartite graph by subdividing a perfect matching. We show that every connected...

We give a (computer assisted) proof that the edges of every graph with maximum degree 3 and girth at least 17 may be 5-colored (possibly improperly) so that the complement of each color class is bipartite. Equivalently, every such graph admits a homomorphism to the Clebsch graph (Fig. 1). Hopkins and Staton [8] and Bondy and Locke [2] proved that every (sub)cubic graph of girth at least 4 has a...

We give a (computer assisted) proof that the edges of every graph with maximum degree 3 and girth at least 17 may be 5-colored (possibly improperly) so that the complement of each color class is bipartite. Equivalently, every such graph admits a homomorphism to the Clebsch graph (Fig. 1). Hopkins and Staton [11] and Bondy and Locke [2] proved that every (sub)cubic graph of girth at least 4 has ...

The Clebsch-Gordan formulas for d-function are studied for both nonunitary finite dimensional representation and unitary infinite dimensional representation of the SU (1, 1) group. It is shown that, in addition to the usual Clebsch-Gordan formulas for the above-mentioned two representations, there exists an additional type of Clebsch-Gordan fornmla connecting the nonunitary and unitary d-functi...

In this article we investigate the structure of a twisted cohomology group of the first homology of a trivalent graph with a coefficient associated with its quantum Clebsch-Gordan condition. By using this investigation we shall show the existence of the external edge cocycles defined in [3] and a finite algorithm to construct such cocycles. Such cocycles are used for a combinatorial realization...

The angular bispectrum of spherical random fields has recently gained an enormous importance, especially in connection with statistical inference on cosmological data. In this paper, we provide expressions for its moments of arbitrary order and we use these results to establish a multivariate central limit theorem and higher order approximations. The results rely upon combinatorial methods from...