Let Γ be a distance-regular graph with diameterd ≥ 3 and heighth = 2, whereh = max{i : pd,i 6= 0}. Suppose that for every α in Γ and every β in Γd(α), the induced subgraph on Γd(α)∩ Γ2(β) is isomorphic to a complete multipartite graph Kt×2 with t ≥ 2. Then d = 4 and Γ is isomorphic to the Johnson graph J(10, 4).

We propose an unsupervised keyphrase extraction model that encodes topical information within a multipartite graph structure. Our model represents keyphrase candidates and topics in a single graph and exploits their mutually reinforcing relationship to improve candidate ranking. We further introduce a novel mechanism to incorporate keyphrase selection preferences into the model. Experiments con...

A cycle in a multipartite graph G is gregarious if it contains at most one vertex from each partite set of G. The complete n-partite graph with partite sets of size m, denoted by Kn(m) is shown to have a decomposition into gregarious 4-cycles. The notion of a gregarious 4-cycle decomposition of this type was introduced in [3]. A 4-cycle decomposition of Kn(m) is circular if it is it is invarian...

Let α be a permutation of the vertex set V (G ) of a connected graph G. Define the total relative displacement of α in G by δα (G )G ∑ x,y∈V(G) dG (x, y)AdG (α (x), α (y)) , where dG (x, y) is the length of the shortest path between x and y in G. Let π*(G ) be the maximum value of δα (G ) among all permutations of V (G ). The permutation which realizes π*(G ) is called a chaotic mapping of G. I...

A digraph obtained by replacing each edge of a complete n-partite (n 2:: 2) graph by an arc or a pair of mutually opposite arcs with the same end vertices is called a semicomplete n-partite digraph or semicomplete multipartite digraph (abbreviated to SMD). In this paper we show the following result for a semicomplete multipartite digraph of order p with the partite sets VI, 112, ... , Vn. Let r...

Let [Formula: see text] be a simple graph. The resistance distance between [Formula: see text], denoted by [Formula: see text], is defined as the net effective resistance between nodes i and j in the corresponding electrical network constructed from G by replacing each edge of G with a resistor of 1 Ohm. The resistance-distance matrix of G, denoted by [Formula: see text], is a [Formula: see tex...

Let be an antipodal distance-regular graph of diameter 4, with eigenvalues θ0 > θ1 > θ2 > θ3 > θ4. Then its Krein parameter q4 11 vanishes precisely when is tight in the sense of Jurišić, Koolen and Terwilliger, and furthermore, precisely when is locally strongly regular with nontrivial eigenvalues p := θ2 and −q := θ3. When this is the case, the intersection parameters of can be parametrized b...

A graph G is arbitrarily partitionable if every sequence (n1, n2, ..., np) of positive integers summing up to |V (G)| is realizable in G, i.e. there exists a partition (V1, V2, ..., Vp) of V (G) such that Vi induces a connected subgraph of G on ni vertices for every i ∈ {1, 2, ..., p}. Given a family F(n) of graphs with order n ≥ 1, a kernel of sequences for F(n) is a set KF (n) of sequences su...

An i-triangulated graph is a graph in which every odd cycle has two non-crossing chords; i-triangulated graphs form a subfamily of perfect graphs. A slightly more general family of perfect graphs are clique-separable graphs. A graph is clique-separable precisely if every induced subgraph either has a clique cutset, or is a complete multipartite graph or a clique joined to an arbitrary bipartite...

In 1970s, Gutman introduced the concept of the energy E (G) for a simple graph G, which is defined as the sum of the absolute values of the eigenvalues of G. This graph invariant has attracted much attention, and many lower and upper bounds have been established for some classes of graphs among which bipartite graphs are of particular interest. But there are only a few graphs attaining the equa...