Yubin Gao

First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph

Let G be a mixed graph and let L(G) be the Laplacian matrix of the graph G. The first eigenvalue and the first eigenvectors of G are respectively referred to the least nonzero eigenvalue and the corresponding eigenvectors of L(G). In this paper we focus on the properties of the first eigenvalue and the first eigenvectors of a nonsingular unicyclic mixed graph (abbreviated to a NUM graph). We introduce the notion of characteristic set associated with the first eigenvectors, and then obtain some results on the sign structure of the first eigenvectors. By these results we determine the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order and fixed girth, and the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order.

Sign patterns with minimum rank 2 and upper bounds on minimum ranks

A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0}. The minimum rank (resp., rational minimum rank) of a sign pattern matrix 𝒜 is the minimum of the ranks of the real (resp., rational) matrices whose entries have signs equal to the corresponding entries of 𝒜. The notion of a condensed sign pattern is introduced. A new, insightful proof of the rational realizability of the minimum rank of a sign pattern with minimum rank 2 is obtained. Several characterizations of sign patterns with minimum rank 2 are established, along with linear upper bounds for the absolute values of an integer matrix achieving the minimum rank 2. A known upper bound for the minimum rank of a (+, −) sign pattern in terms of the maximum number of sign changes in the rows of the sign pattern is substantially extended to obtain upper bounds for the rational minimum ranks of general sign pattern matrices. The new concept of the number of polynomial sign changes of a sign vector is crucial for this extension. Another known upper bound for the minimum rank of a (+, −) sign pattern in terms of the smallest number of sign changes in the rows of the sign pattern is also extended to all sign patterns using the notion of the number of strict sign changes. Some examples and open problems are also presented.

Bounds on the local bases of primitive nonpowerful nearly reducible sign patterns

In this work, we study the kth local base, which is a generalization of the base, of a primitive non-powerful nearly reducible sign pattern of order n ≥ 7. We obtain the sharp bound together with a complete characterization of the equality case, of the kth local bases for primitive non-powerful nearly reducible sign patterns. We also show that there exist “gaps” in the kth local base set of primitive non-powerful nearly reducible sign patterns.

The kth upper and lower bases of primitive nonpowerful minimally strong signed digraphs

In this article, we study the kth upper and lower bases of primitive nonpowerful minimally strong signed digraphs. A bound on the kth upper bases for primitive nonpowerful minimally strong signed digraphs is obtained, and the equality case of the bound is characterized. For the kth lower bases, we obtain some bounds. For some cases, the bounds are best possible and the extremal signed digraphs are characterized. We also show that there exist ‘gaps’ in both the kth upper base set and the kth lower base set of primitive nonpowerful minimally strong signed digraphs.

SPECTRALLY ARBITRARY COMPLEX SIGN PATTERN MATRICES

An n × n complex sign pattern matrix S is said to be spectrallyarbitraryif for everymonic nth degree polynomial f (λ) with coefficients from C, there is a complex matrix in the complex sign pattern class of S such that its characteristic polynomial is f (λ). If S is a spectrally arbitrarycomplex sign pattern matrix, and no proper subpattern of S is spectrallyarbitrary , then S is a minimal spectrallyarbitrarycomplex sign pattern matrix. This paper extends the Nilpotent- Jacobian method for sign pattern matrices to complex sign pattern matrices, establishing a means to show that an irreducible complex sign pattern matrix and all its superpatterns are spectrally arbitrary. This method is then applied to prove that for every n ≥ 2t here exists ann × n irreducible, spectrallyarbitrarycomplex sign pattern with exactly3 n nonzero entries. In addition, it is shown that every n × n irreducible, spectrallyarbitrarycomplex sign pattern matrix has at least 3 n − 1 nonzero entries.

On the exponents of two-colored digraphs with two cycles†

We consider the special primitive two-colored digraphs whose uncolored digraph has n+s vertices and consists of one n-cycle and one n−t-cycle. We give some primitivity conditions and an upper bound on the exponent. Further, for the case s=0, we give a tight upper bound on the exponent and the characterization of extremal two-colored digraphs. †Research supported by NNSF of China (No. 10571163) and NSF of Shanxi (No. 20041010).

Essential sign change numbers of full sign pattern matrices

Abstract A sign pattern (matrix) is a matrix whose entries are from the set {+, −, 0} and a sign vector is a vector whose entries are from the set {+, −, 0}. A sign pattern or sign vector is full if it does not contain any zero entries. The minimum rank of a sign pattern matrix A is the minimum of the ranks of the real matrices whose entries have signs equal to the corresponding entries of A. The notions of essential row sign change number and essential column sign change number are introduced for full sign patterns and condensed sign patterns. By inspecting the sign vectors realized by a list of real polynomials in one variable, a lower bound on the essential row and column sign change numbers is obtained. Using point-line confiurations on the plane, it is shown that even for full sign patterns with minimum rank 3, the essential row and column sign change numbers can differ greatly and can be much bigger than the minimum rank. Some open problems concerning square full sign patterns with large minimum ranks are discussed.

Sign patterns that allow diagonalizability revisited

Characterization of sign patterns that allow diagonalizability has been a long-standing open problem. In this article, necessary and sufficient conditions for a sign pattern to allow diagonalizability are obtained, in terms of allowing related properties. Some properties of normal sign patterns are considered. In particular, it is shown that normal sign patterns of order up to 3 allow diagonalizability. Two combinatorial necessary conditions for a sign pattern to allow diagonalizability are also presented.

The m-competition indices of symmetric primitive digraphs without loops

For positive integers m and n with 1 � mn, the m-competition index (generalized competition index) of a primitive digraph D of order n is the smallest positive integer k such that for every pair of vertices x and y in D, there exist m distinct vertices v1,v2,...,vm such that there exist walks of length k from x to vi and from y to vi for each i = 1,...,m. In this paper, we study the generalized competition indices of symmetric primitive digraphs without loops. We determine the generalized competition index set and characterize the digraphs in this class with largest generalized competition index.

Exponents of 2-colorings of loopless, symmetric digraphs

A 2-coloring (G 1, G 2) is primitive if there exist non-negative integers h and k with h + k > 0 such that for each pair (u, v) of vertices there exists an (h, k)-walk in G from u to v. The exponent of (G 1, G 2) is the minimum value of h + k taken over all such h and k. In this article, we consider 2-colorings of loopless, symmetric digraphs give the conditions for a 2-coloring of loopless, symmetric digraph to be primitive and establish an upper bound on the exponents.