In this paper, we study the characteristic polynomials of the line graphs of generalized Bethe trees. We give an infinite family of such graphs sharing the same smallest eigenvalue. Our family generalizes the family of coronas of complete graphs discovered by Cvetkovic and Stevanovic.
In this paper, we give a combinatorial characterization of the special graphs of fat Hoffman graphs containing $mathfrak{K}_{1,2}$ with smallest eigenvalue greater than -3, where $mathfrak{K}_{1,2}$ is the Hoffman graph having one slim vertex and two fat vertices.
In this paper, we show that all fat Hoffman graphs with smallest eigenvalue at least -1-tau, where tau is the golden ratio, can be described by a finite set of fat (-1-tau)-irreducible Hoffman graphs. In the terminology of Woo and Neumaier, we mean t
hat every fat Hoffman graph with smallest eigenvalue at least -1-tau is an H-line graph, where H is the set of isomorphism classes of maximal fat (-1-tau)-irreducible Hoffman graphs. It turns out that there are 37 fat (-1-tau)-irreducible Hoffman graphs, up to isomorphism.
Since Cho and Kim (2005) showed that the competition graph of a doubly partial order is an interval graph, it has been actively studied whether or not the same phenomenon occurs for other variants of competition graph and interesting results have bee
n obtained. Continuing in the same spirit, we study the competition hypergraph, an interesting variant of the competition graph, of a doubly partial order. Though it turns out that the competition hypergraph of a doubly partial order is not always interval, we completely characterize the competition hypergraphs of doubly partial orders which are interval.
In this paper, we classify the connected non-bipartite integral graphs with spectral radius three.
