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Lihua You

Publications and source records attributed to Lihua You.

49 records · Page 3Linked to original sources
arXiv2016

On some properties of three different types of triangular blocked tensors

We define three types of upper (and lower) triangular blocked tensors, which are all generalizations of the triangular blocked matrices. We study some basic properties and characterizations of these three types of triangular blocked tensors. We obtain the formulas for the determinants, characteristic polynomials and spectra of the first and second type triangular blocked tensors, and give an example to show that these formulas no longer hold for the third type triangular blocked tensors. We prove that the product of any two $(n_1,\cdots,n_r)$-upper (or lower) triangular blocked tensors of the first or second or third type is still an $(n_1,\cdots,n_r)$-upper (or lower) triangular blocked tensor of the same type. We also prove that, if an $(n_1,\cdots,n_r)$-upper triangular blocked tensor of the first or second or third type has a left $k$-inverse, then its unique left $k$-inverse is still an $(n_1,\cdots,n_r)$-upper triangular blocked tensor of the same type. Also if it has a right $k$-inverse, then all of its right $k$-inverses are still $(n_1,\cdots,n_r)$-upper triangular blocked tensors of the same type. By showing that the left $k$-inverse (if any) of a weakly irreducible nonsingular $M$-tensor is a positive tensor, we show that the left $k$-inverse (if any) of a first or second or third type canonical $(n_1,\cdots,n_r)$-upper triangular blocked nonsingular $M$-tensor is an $(n_1,\cdots,n_r)$-upper triangular blocked tensor of the same type all of whose diagonal blocks are positive tensors. We also show that every order $m$ dimension $n$ tensor is permutation similar to some third type normal upper triangular blocked tensor (all of whose diagonal blocks are irreducible). We give an example to show that this is not true for the first type canonical upper triangular blocked tensor.

math.RA↗
arXiv2015

On a conjecture for the signless Laplacian spectral radius of cacti with given matching number

A connected graph $G$ is a cactus if any two of its cycles have at most one common vertex. Let $\ell_n^m$ be the set of cacti on $n$ vertices with matching number $m.$ S.C. Li and M.J. Zhang determined the unique graph with the maximum signless Laplacian spectral radius among all cacti in $\ell_n^m$ with $n=2m$. In this paper, we characterize the case $n\geq 2m+1$. This confirms the conjecture of Li and Zhang(S.C. Li, M.J. Zhang, On the signless Laplacian index of cacti with a given number of pendant vetices, Linear Algebra Appl. 436, 2012, 4400--4411). Further, we characterize the unique graph with the maximum signless Laplacian spectral radius among all cacti on $n$ vertices.

math.CO↗
arXiv2015

Sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix

In this paper, we obtain the sharp upper and lower bounds for the spectral radius of a nonnegative irreducible matrix. We also apply these bounds to various matrices associated with a graph or a digraph, obtain some new results or known results about various spectral radii, including the adjacency spectral radius, the signless Laplacian spectral radius, the distance spectral radius, the distance signless Laplacian spectral radius of a graph or a digraph.

math.CO↗
arXiv2014

Some remarks on P, P_0, B and B_0 tensors

Recently, Song and Qi extended the concept of P, P_0 and B matrices to P, P_0, B and B_0 tensors, obtained some properties about these tensors, and proposed many questions for further research. In this paper, we answer three questions mentioned as above and obtain further results about P, P_0, B and B_0 tensors.

math.CO↗
arXiv2014

On the exponent set of nonnegative primitive tensors

In this paper, we present a necessary and sufficient condition for a nonnegative tensor to be a primitive one, show that the exponent set of nonnegative primitive tensors with order $m(\ge n)$ and dimension $n$ is $\{k| 1\le k\le (n-1)^2+1\}. $

math.CO↗
arXiv2014

The Minimal Total Irregularity of Graphs

In \cite{2012a}, Abdo and Dimitov defined the total irregularity of a graph $G=(V,E)$ as \hskip3.3cm $\rm irr_{t}$$(G) = \frac{1}{2}\sum_{u,v\in V}|d_{G}(u)-d_{G}(v)|, $ \noindent where $d_{G}(u)$ denotes the vertex degree of a vertex $u\in V$. In this paper, we investigate the minimal total irregularity of the connected graphs, determine the minimal, the second minimal, the third minimal total irregularity of trees, unicyclic graphs, bicyclic graphs on $n$ vertices, and propose an open problem for further research.

math.CO↗
arXiv2014

Spectral radius and signless Laplacian spectral radius of strongly connected digraphs

Let D be a strongly connected digraph and A(D) be the adjacency matrix of D. Let diag(D) be the diagonal matrix with outdegrees of the vertices of D and Q(D) = diag(D) + A(D) be the signless Laplacian matrix of D. The spectral radius of Q(D) is called the signless Laplacian spectral radius of D, denoted by q(D). In this paper, we give sharp bound on q(D) with outdegree sequence and compare the bound with some known bounds, establish some sharp upper or lower bound on q(D) with some given parameter such as clique number, girth or vertex connectivity, and characterize the corresponding extremal digraph or proposed open problem. In addition, we also determine the unique digraph which achieves the minimum (or maximum), the second minimum (or maximum), the third minimum, the fourth minimum spectral radius and signless Laplacian spectral radius among all strongly connected digraphs, and answer the open problem proposed by Lin-Shu [H.Q. Lin, J.L. Shu, A note on the spectral characterization of strongly connected bicyclic digraphs, Linear Algebra Appl. 436 (2012) 2524{2530].

math.CO↗
arXiv2013

On the similarity of Tensors

Let $\mathbb{P}_n$ be the set of all matrices which have the same zero patterns with some permutation matrix of order $n$. In this paper, we prove the following result: Let $\mathbb{I}$ be the unit tensor of order $m\ge3$ and dimension $n\ge2$. Suppose that $P$ and $Q$ are two matrices with $P\mathbb{I}Q=\mathbb{I}$, then $P,Q\in \mathbb{P}_n$. This gives a characterization for the similarities of tensors with order $m\ge3$.

math.CO↗
arXiv2013

A conjecture on the primitive degree of Tensors

In this paper, we prove: Let A be a nonnegative primitive tensor with order m and dimension n. Then its primitive degree R(A)\leq (n-1)^2+1, and the upper bound is sharp. This confirms a conjecture of Shao [7].

math.CO↗
arXiv2013

Some sharp bounds on the distance signless Laplacian spectral radius of graphs

M. Aouchiche and P. Hansen proposed the distance Laplacian and the distance signless Laplacian of a connected graph [Two Laplacians for the distance matrix of a graph, LAA 439 (2013) 21{33]. In this paper, we obtain three theorems on the sharp upper bounds of the spectral radius of a nonnegative matrix, then apply these theorems to signless Laplacian matrices and the distance signless Laplacian matrices to obtain some sharp bounds on the spectral radius, respectively. We also proposed a known result about the sharp bound of the signless Laplacian spectral radius has a defect.

math.CO↗
arXiv2013

Notes on the sum of powers of the signless Laplacian eigenvalues of graphs

For a graph $G$ and a non-zero real number $α$, the graph invariant $S_α(G)$ is the sum of the $α^{th}$ power of the non-zero signless Laplacian eigenvalues of $G$. In this paper, we obtain the sharp bounds of $S_α(G)$ for a connected bipartite graph $G$ on $n$ vertices and a connected graph $G$ on $n$ vertices having a connectivity less than or equal to $k$, respectively, and propose some open problems for future research.

math.CO↗
arXiv2013

On a conjecture for the signless Laplacian eigenvalues

Let $G$ be a simple graph with $n$ vertices and $e(G)$ edges, and $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)\geq0$ be the signless Laplacian eigenvalues of $G.$ Let $S_k^+(G)=\sum_{i=1}^{k}q_i(G),$ where $k=1, 2, \ldots, n.$ F. Ashraf et al. conjectured that $S_k^+(G)\leq e(G)+\binom{k+1}{2}$ for $k=1, 2, \ldots, n.$ In this paper, we give various upper bounds for $S_k^+(G),$ and prove that this conjecture is true for the following cases: connected graph with sufficiently large $k,$ unicyclic graphs and bicyclic graphs for all $k,$ and tricyclic graphs when $k\neq 3.$

math.CO↗