Guanglong Yu

Spectral radius of strongly connected digraphs

Let D be a digraph with vertex set V ( D ) and A be the adjacency matrix of D . The largest eigenvalue of A , denoted by ? ( D ) , is called the spectral radius of the digraph D . In this paper, we establish some sharp upper or lower bounds for digraphs with some given graph parameters, such as clique number, girth, and vertex connectivity, and characterize the corresponding extremal graphs. In addition, we give the exact value of the spectral radii of those digraphs.

The base of a primitive, nonpowerful sign pattern with exactly d nonzero diagonal entries

In [J.Y. Shao, L.H. You, H.Y. Shan, Bound on the base of irreducible generalized sign pattern matrices, Linear Algebra Appl. 427 (2007) 2-3, 285-300], Shao, You and Shan extended the concept of the base from powerful sign pattern matrices to nonpowerful (generalized) sign pattern matrices. It is well known that the properties of the power sequences of different classes of sign pattern matrices may be very different. In this paper, we consider the base set of the primitive nonpowerful square sign pattern matrices of order n with exactly d (with d>=1) nonzero diagonal entries. The base set is shown to be {2,3,...,3n-d-1}. The extremal sign pattern matrices with both the least number n+d nonzero entries and the maximum base 3n-d-1 are characterized.

Gaps in the base set of primitive nonpowerful sign patterns

For a primitive nonpowerful square sign pattern A, the base of A, denoted by l(A), is the least positive integer l such that every entry of A l is #. In this article, we consider the base set of the primitive nonpowerful sign pattern matrices. Some useful results about the bases for the sign pattern matrices are presented there. Some special sign pattern matrices with given bases are characterized and more ‘gaps’ in the base set are shown.

BOUNDS OF SPECTRAL RADII OF K2,3-MINOR FREE GRAPHS

Let A(G) be the adjacency matrix of a graph G. The largest eigenvalue of A(G) is called spectral radius of G. In this paper, an upper bound of spectral radii of K2,3-minor free graphs with order n is shown to be 3 + r n 7 . In order to prove this upper bound, a structural characterization of K2,3-minor free graphs is presented in this paper.

Bounds of spectral radii on edge-most outer-planar bipartite graphs

Let A(G) be the adjacency matrix of a graph G. The largest eigenvalue of A(G) is called the spectral radius of the graph G. For an outer-planar bipartite graph G with an order n and m(G) edges, if n ≥ 2 is even and m(G) =  , or n ≥ 3 is odd and m(G) =  , G is called the edge-most. For the spectral radius of an edge-most outer-planar bipartite graph G with an order n, two upper bounds are shown to be that: if n is even, then ; if n is odd, then .