Showing papers by "Xueliang Li published in 2014"
TL;DR: Upper bounds of the rainbow connection number of outerplanar graphs with small diameters are given to prove that rc ( G ) ⩽ 5 if G is a bridgeless graph with diameter 2, while rc (G ) ⦽ 9 if G are a bridGeless graphs with diameter 3.
24 citations
19 Dec 2014
TL;DR: It is proved that for any two connected graphs G and H, κ3(G^H)≥ κ 3(G)|V(H)|, and the upper bounds are sharp.
Abstract: The generalized k-connectivity κk(G) of a graph G, first introduced by Hager, is a natural generalization of the concept of (vertex-)connectivity. Denote by G^H and GBH the lexicographic product and Cartesian product of two graphs G and H, respectively. In this paper, we prove that for any two connected graphs G and H, κ3(G^H)≥ κ3(G)|V(H)|. We also give upper bounds for κ3(GB H) and κ3(G^H). Moreover, all the bounds are sharp.
21 citations
TL;DR: In this paper, the concept of general Randi'c matrix has been introduced and lower and upper bounds for the general Randic spectral radius of a connected graph have been obtained.
Abstract: Let $G$ be a simple graph with vertex set $V(G) = {v_1, v_2,ldots, v_n}$ and $d_i$ the degree of its vertex $v_i$, $i = 1, 2, dots, n$. Inspired by the Randic matrix and the general Randic index of a graph, we introduce the concept of general Randi'c matrix $textbf{R}_alpha$ of $G$, which is defined by $(textbf{R}_alpha)_{i,j}=(d_id_j)^alpha$ if $v_i$ and $v_j$ are adjacent, and zero otherwise. Similarly, the general Randic eigenvalues are the eigenvalues of the general Randic} matrix, the greatest general Randic eigenvalue is the general Randic spectral radius of $G$, and the general Randic energy is the sum of the absolute values of the general Randic eigenvalues. In this paper, we prove some properties of the general Randi'c matrix and obtain lower and upper bounds for general Randic energy, also, we get some lower bounds for general Randic spectral radius of a connected graph. Moreover, we give a new sharp upper bound for the general Randic energy when $alpha=-1/2$.
17 citations
TL;DR: The rainbow vertex-connection number, r v c ( G ) is the minimum number of colors needed to color its vertices such that every pair of vertices is connected by at least one path whose internal vertices have distinct colors.
15 citations
TL;DR: In this paper, the authors deduce an integral formula for the skew energy of an oriented graph and determine all oriented graphs with minimal skew energy among all connected oriented graphs on n vertices with m (n ≤ m < 2(n − 2)) arcs, which is an analogy to the conjecture for the energy of undirected graphs proposed by Caporossi et al.
Abstract: Let S(G � ) be the skew-adjacency matrix of an oriented graph G � . The skew energy of Gis defined as the sum of all singular values of its skew- adjacency matrix S(G � ). In this paper, we first deduce an integral formula for the skew energy of an oriented graph. Then we determine all oriented graphs with minimal skew energy among all connected oriented graphs on n vertices with m (n ≤ m < 2(n − 2)) arcs, which is an analogy to the conjecture for the energy of undirected graphs proposed by Caporossi et al. (G. Caporossi, D. Cvetkovic, I. Gutman, P. Hansen, Variable neighborhood search for extremal graphs. 2. Finding graphs with external energy, J. Chem. Inf. Comput. Sci. 39 (1999) 984-996.).
9 citations
TL;DR: In this article, the Randi'c incidence matrix of a simple graph is introduced, denoted by the singular values of the I_R(G) matrix, which is defined as the $ntimes m$ matrix whose entry is $(d_i)^{-frac{1}{2}}$ if $v_i$ is incident to $e_j$ and $0$ otherwise.
Abstract: Let $G$ be a simple graph with vertex set $V(G) = {v_1, v_2,ldots, v_n}$ and edge set $E(G) = {e_1, e_2,ldots, e_m}$ Similar to the Randi'c matrix, here we introduce the Randi'c incidence matrix of a graph $G$, denoted by $I_R(G)$, which is defined as the $ntimes m$ matrix whose $(i,j)$-entry is $(d_i)^{-frac{1}{2}}$ if $v_i$ is incident to $e_j$ and $0$ otherwise Naturally, the Randi'c incidence energy $I_RE$ of $G$ is the sum of the singular values of $I_R(G)$ We establish lower and upper bounds for the Randic incidence energy Graphs for which these bounds are best possible are characterized Moreover, we investigate the relation between the Randic incidence energy of a graph and that of its subgraphs Also we give a sharp upper bound for the Randic incidence energy of a bipartite graph and determine the trees with the maximum Randic incidence energy among all $n$-vertex trees As a result, some results are very different from those for incidence energy
7 citations
01 Jan 2014
TL;DR: In this paper, it was shown that G is not a tree if and only if rc(G) ≤ m − 2, where m is the number of edges of the graph.
Abstract: A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. A nontrivial connected graph G is rainbow connected if there is a rainbow path connecting any two vertices, and the rainbow connection number of G, denoted by rc(G), is the minimum number of colors that are needed in order to make G rainbow connected. Chartrand et al. showed that G is a tree if and only if rc(G )= m, and it is easy to see that G is not a tree if and only if rc(G) ≤ m − 2, where m is the number of edges of G. So an interesting problem arises: Characterize the graphs G with rc(G )= m − 2. In this paper, we resolve this problem. Furthermore, we also characterize the graphs G with rc(G )= m − 3.
4 citations
TL;DR: In this paper, the minimal skew energy of a digraph with skew-adjacency matrix is defined as the sum of the norms of all the eigenvalues of the skew matrix.
Abstract: Let $D$ be a digraph with skew-adjacency matrix $S(D)$. Then the skew energy of $D$ is defined as the sum of the norms of all eigenvalues of $S(D)$. Denote by $mathcal{O}_n$ the class of digraphs of order $n$ with no even cycles, and by $mathcal{O}_{n,m}$ the class of digraphs in $mathcal{O}_n$ with $m$ arcs. In this paper, we first give the minimal skew energy digraphs in $mathcal{O}_n$ and $mathcal{O}_{n,m}$ with $n-1leq mleq frac{3}{2}(n-1)$. Then we determine the maximal skew energy digraphs in $mathcal{O}_{n,n}$ and $mathcal{O}_{n,n+1}$, and in the latter case we assume that $n$ is even.
4 citations
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TL;DR: This article proves accurate connections (inequalities) between generalized graph entropies, distinct graph energies and topological indices, and obtains some extremal properties of nine generalized graphEntropies by employing distinct graph Energy and Topological indices.
Abstract: The entropy of a graph is an information-theoretic quantity which expresses the complexity of a graph \cite{DM1,M}. After Shannon introduced the definition of entropy to information and communication, many generalizations of the entropy measure have been proposed, such as R\'enyi entropy and Dar\`oczy's entropy. In this article, we prove accurate connections (inequalities) between generalized graph entropies, distinct graph energies and topological indices. Additionally, we obtain some extremal properties of nine generalized graph entropies by employing distinct graph energies and topological indices.
3 citations
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TL;DR: In this paper, the authors defined a weighted skew adjacency matrix with Randi weight, the skew Randi-c matrix, of a simple graph with an orientation σ, which assigns to each edge a direction so that σ becomes a directed graph.
Abstract: Let $G$ be a simple graph with an orientation $\sigma$, which assigns to each edge a direction so that $G^\sigma$ becomes a directed graph. $G$ is said to be the underlying graph of the directed graph $G^\sigma$. In this paper, we define a weighted skew adjacency matrix with Rand\'c weight, the skew Randi\'c matrix ${\bf R_S}(G^\sigma)$, of $G^\sigma$ as the real skew symmetric matrix $[(r_s)_{ij}]$ where $(r_s)_{ij} = (d_id_j)^{-\frac{1}{2}}$ and $(r_s)_{ji} = -(d_id_j)^{-\frac{1}{2}}$ if $v_i \rightarrow v_j$ is an arc of $G^\sigma$, otherwise $(r_s)_{ij} = (r_s)_{ji} = 0$. We derive some properties of the skew Randi\'c energy of an oriented graph. Most properties are similar to those for the skew energy of oriented graphs. But, surprisingly, the extremal oriented graphs with maximum or minimum skew Randi\'c energy are completely different.
3 citations
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TL;DR: This paper introduces a new method of searching (proposed) optimum additive codes from circulant graphs based on Danielsen and Parker's proved that every self-dual additive code over GF(4) is equivalent to a graph code.
Abstract: In 2006, Danielsen and Parker [8] proved that every self-dual additive code over GF(4) is equivalent to a graph code. So, graph is an important tool for searching (proposed) optimum codes. In this paper, we introduce a new method of searching (proposed) optimum additive codes from circulant graphs.
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TL;DR: In this paper, the Randi´c incidence matrix of a simple graph is introduced, denoted by $I_R(G), which is defined as the $n\times m$ matrix whose $(i, j)$-entry is $(d_i)^{-\frac{1}{2}}$ if $v_i$ is incident to $e_j$ and $0$ otherwise.
Abstract: Let $G$ be a simple graph with vertex set $V(G) = \{v_1, v_2,\ldots, v_n\}$ and edge set $E(G) = \{e_1, e_2,\ldots, e_m\}$. Similar to the Randi\'c matrix, here we introduce the Randi\'c incidence matrix of a graph $G$, denoted by $I_R(G)$, which is defined as the $n\times m$ matrix whose $(i, j)$-entry is $(d_i)^{-\frac{1}{2}}$ if $v_i$ is incident to $e_j$ and $0$ otherwise. Naturally, the Randi\'c incidence energy $I_RE$ of $G$ is the sum of the singular values of $I_R(G)$. We establish lower and upper bounds for the Randi\'c incidence energy. Graphs for which these bounds are best possible are characterized. Moreover, we investigate the relation between the Randi\'c incidence energy of a graph and that of its subgraphs. Also we give a sharp upper bound for the Randi\'c incidence energy of a bipartite graph and determine the trees with the maximum Randi\'c incidence energy among all $n$-vertex trees. As a result, some results are very different from those for incidence energy.
01 Jan 2014
TL;DR: In this paper, a graph with k distinct eigenvalues with respect to the adjacency and (normalized) Laplacian matrix is studied. And the authors give an algebraic characterization of the graph with the same eigenvalue.
Abstract: We rst investigate the Hermitian matrices with k distinct eigenvalues, and then give an algebraic characterization to a graph with k distinct eigenvalues with respect to the adjacency and (normalized) Laplacian matrix.
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TL;DR: In this article, it was shown that for some constant k and p > K logn/n, w.h.p. is tripartite and it does not hold when p = 0.
Abstract: A sparse version of Mantel's Theorem is that, for sufficientlylarge p, with high probability (w.h.p.), every maximum triangle-free subgraph of G(n,p) is bipartite. DeMarco and Kahn proved this for p > K p logn/n for some con- stant K, and apart from the value of the constant, this bound is the best possible. Denote by T3 the 3-uniform hypergraph with vertex set {a,b,c,d,e} and edge set {abc,ade,bde}. Frankl and Furedi showed that the maximum 3-uniform hypergraph on n vertices containing no copy of T3 is tripartite for n > 3000. For some integer k, let G k (n,p) be the random k-uniform hyper- graph. Balogh et al. proved that for p > K logn/n for some constant K, every maximum T3-free subhypergraph of G 3 (n,p) w.h.p. is tripartite and it does not hold when p = 0.1 √ logn/n. Denote by T4 the 4-uniform hypergraph with vertex set {1,2,3,4,5,6,7} and edge set {1234,1235,4567}. Pikhurko proved that there is an n0 such that for all n ≥ n0, the maximum 4-uniform hyper- graph on n vertices containing no copy of T4 is 4-partite. In this paper, we extend this type of extremal problem in random 4-uniform hypergraphs. We show that for some constant K and p > K logn/n, w.h.p. every maximum T4-free subhypergraph of G 4 (n,p) is 4-partite.
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TL;DR: Two kinds of Erdős–Gallai-type problems for mc(G) are studied, and two of them are completely solve.
Abstract: A path in an edge-colored graph is called a \emph{monochromatic path} if all the edges on the path are colored the same. An edge-coloring of $G$ is a \emph{monochromatic connection coloring} (MC-coloring, for short) if there is a monochromatic path joining any two vertices in $G$. The \emph{monochromatic connection number}, denoted by $mc(G)$, is defined to be the maximum number of colors used in an MC-coloring of a graph $G$. These concepts were introduced by Caro and Yuster, and they got some nice results. In this paper, we will study two kinds of Erd\H{o}s-Gallai-type problems for $mc(G)$, and completely solve them.
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TL;DR: Guran et al. as discussed by the authors introduced the concept of the maximum number of colors used in an MC-coloring of a connected graph, which is called monochromatic-connec-tioncoloring (MCcoloring), introduced by Caro and Yuster.
Abstract: Center for Combinatorics and LPMC-TJKLCNankai University, Tianjin 300071, PR ChinaEmail: guran323@163com, lxl@nankaieducn, qinzhongmei90@163comAbstractAn edge-coloring of a connected graph Gis called a monochromaticconnec-tioncoloring(MC-coloring, for short), introduced by Caro and Yuster, if thereis a monochromatic path joining any two vertices of the graph G Let mc(G)denote the maximum number of colors used in an MC-coloring of a graph GNote that an MC-coloring does not exist if Gis not connected, and in this casewe simply let mc(G) = 0 We use G(n,p) to denote the Erd¨os-R´enyi randomgraph model, in which each of the
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TL;DR: A new method of searching (proposed) optimum formally self-duallinear binary codes from circulant graphs from circULant graphs is introduced.
Abstract: ∗Xueliang Li, Yaping Mao, Meiqin WeiNankai University, Tianjin 300071, ChinaCenter for Combinatorics and LPMC-TJKLCE-mails: lxl@nankai.edu.cn; maoyaping@ymail.com; weimeiqin@mail.nankai.edu.cnandRuihu LiThe air force engineering UniversityInstitute of science, Xi’an 710051, ChinaE-mail: liruihu@aliyun.comAbstractIn 2002, Tonchev first constructed some linear binary codes defined by the adjacencymatrices of undirected graphs. So graph is an important tool for searching optimum code.In this paper, we introduce a new method of searching (proposed) optimum formally self-duallinear binary codes from circulant graphs.AMS Subject Classification 2010: 94B05, 05C50, 05C25.
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TL;DR: In this paper, a necessary and sufficient condition for an Hermitian matrix with simple spectral radius and distinct eigenvalues is given for which graphs have distinct adjacency eigen values.
Abstract: Harary and Schwenk posed the problem forty years ago: Which graphs have distinct adjacency eigenvalues? In this paper, we obtain a necessary and sufficient condition for an Hermitian matrix with simple spectral radius and distinct eigenvalues. As its application, we give an algebraic characterization to the Harary-Schwenk's problem. As an extension of their problem, we also obtain a necessary and sufficient condition for a positive semidefinite matrix with simple least eigenvalue and distinct eigenvalues, which can provide an algebraic characterization to their problem with respect to the (normalized) Laplacian matrix.