Benjian Lv

On the fractional metric dimension of graphs

In Arumugam et al. (2013), Arumugam et al. studied the fractional metric dimension of the Cartesian product of two graphs, and proposed four open problems. In this paper, we determine the fractional metric dimension of vertex-transitive graphs, in particular, the fractional metric dimension of a vertex-transitive, distance-regular graph is expressed in terms of its intersection numbers. As an application, we calculate the fractional metric dimension of Hamming graphs and Johnson graphs, respectively. Moreover, we give an inequality for metric dimension and fractional metric dimension of an arbitrary graph, and determine all graphs for which the equality holds. Finally, we establish bounds on the fractional metric dimension of the Cartesian product of graphs. As a result, we completely solve the four open problems.

The endomorphisms of Grassmann graphs

A graph G is a core if every endomorphism of G is an automorphism. A graph is called a pseudo-core if every its endomorphism is either an automorphism or a colouring. Suppose that J q ( n ,u2006 m ) is a Grassmann graph over a finite field with q elements. We show that every Grassmann graph is a pseudo-core. Moreover, J 2 (4,u20062) is not a core and J q (2 k u2005+u20051,u20062) ( k u2004≥u20042 ) is a core.

More on the Terwilliger algebra of Johnson schemes

Abstract In Levstein and Maldonado (2007), the Terwilliger algebra of the Johnson scheme J ( n , d ) was determined when n ≥ 3 d . In this paper, we determine the Terwilliger algebra T for the remaining case 2 d ≤ n 3 d .

The Terwilliger algebra of Odd graphs

Abstract In [F. Levstein, C. Maldonado, The Terwilliger algebra of the Johnson schemes, Discrete Math. 307 (2007) 1621–1635], Levstein and Maldonado computed the Terwilliger algebra of the Johnson scheme J ( n , m ) when 3 m ≤ n . The distance- m graph of J ( 2 m + 1 , m ) is the Odd graph O m + 1 . In this paper, we determine the Terwilliger algebra of O m + 1 and give its basis.

The eigenvalues of q-Kneser graphs

In this note, by proving some combinatorial identities, we obtain a simple form for the eigenvalues of q-Kneser graphs.

The Terwilliger algebra of the incidence graphs of Johnson geometry, II

Let J ( n , m , m + 1 ) denote the incidence graph of Johnson geometry. It is well known that the Johnson graph J ( n , m ) is a halved graph of J ( n , m , m + 1 ) . Let T and T be the Terwilliger algebra of J ( n , m , m + 1 ) and J ( n , m ) , respectively. In this paper, we focus on the structures of irreducible T -modules, and then completely determine T . Furthermore, we discover the relationship between irreducible modules of T and T . As a result, the algebra T is determined again.

Endomorphisms of Twisted Grassmann Graphs

A graph G is called a pseudo-core if every endomorphism of G is either an automorphism or a colouring. An interesting problem in graph theory is to distinguish whether a graph is a core. The twisted Grassmann graphs, constructed by van Dam and Koolen in (Invent Math 162:189–193, 2005), are the first known family of non-vertex-transitive distance-regular graphs with unbounded diameter. In this paper, we show that every twisted Grassmann graph is a pseudo-core.

Error-tolerance pooling designs based on Johnson graphs

As an application of the new model for pooling designs proposed by the last two authors in Guo and Wang (J Combin Theory Ser A 118:2056–2058, 2011), we construct a family of pooling designs based on the

Two lower bounds for generalized 3-connectivity of Cartesian product graphs

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