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Conflict-free (vertex)-connection numbers of graphs with diameters 2, 3 and 4Feb 28 2019A path in an(a) edge(vertex)-colored graph is called a conflict-free path if there exists a color used on only one of its edges(vertices). An(A) edge(vertex)-colored graph is called conflict-free (vertex-)connected if for each pair of distinct vertices, ... More

Derivatives and real roots of graph polynomialsJan 08 2016Jan 12 2016Graph polynomials are polynomials assigned to graphs. Interestingly, they also arise in many areas outside graph theory as well. Many properties of graph polynomials have been widely studied. In this paper, we survey some results on the derivative and ... More

Randić energy and Randić eigenvaluesApr 22 2014Apr 23 2014Let $G$ be a graph of order $n$, and $d_i$ the degree of a vertex $v_i$ of $G$. The Randi\'c matrix ${\bf R}=(r_{ij})$ of $G$ is defined by $r_{ij} = 1 / \sqrt{d_jd_j}$ if the vertices $v_i$ and $v_j$ are adjacent in $G$ and $r_{ij}=0$ otherwise. The ... More

The skew-rank of oriented graphsApr 29 2014An oriented graph $G^\sigma$ is a digraph without loops and multiple arcs, where $G$ is called the underlying graph of $G^\sigma$. Let $S(G^\sigma)$ denote the skew-adjacency matrix of $G^\sigma$. The rank of the skew-adjacency matrix of $G^\sigma$ is ... More

Solutions for two conjectures on kaleidoscopic edge-coloringsNov 24 2016For an $r$-regular graph $G$, we define an edge-coloring $c$ with colors from $\{1,2,\cdots,$ $k\}$, in such a way that any vertex of $G$ is incident to at least one edge of each color. The multiset-color $c_m(v)$ of a vertex $v$ is defined as the ordered ... More

New versions of the all-ones problemDec 01 2005We study three new versions of the All-Ones Problem and the Minimum All-Ones Problem. The original All-Ones Problem is simply called the Vertex-Vertex Problem, and the three new versions are called the Vertex-Edge Problem, the Edge-Vertex Problem and ... More

A survey on the skew energy of oriented graphsApr 21 2013May 18 2015Let $G$ be a simple undirected graph with adjacency matrix $A(G)$. The energy of $G$ is defined as the sum of absolute values of all eigenvalues of $A(G)$, which was introduced by Gutman in 1970s. Since graph energy has important chemical applications, ... More

Skew-spectra and skew energy of various products of graphsMay 31 2013Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. Then the spectrum of $S(G^\sigma)$ consisting of all the eigenvalues of $S(G^\sigma)$ is called the skew-spectrum of $G^\sigma$, ... More

On the maximal energy tree with two maximum degree verticesMar 20 2011Mar 29 2011For a simple graph $G$, the energy $E(G)$ is defined as the sum of the absolute values of all eigenvalues of its adjacent matrix. For $\Delta\geq 3$ and $t\geq 3$, denote by $T_a(\Delta,t)$ (or simply $T_a$) the tree formed from a path $P_t$ on $t$ vertices ... More

The energy of random graphsSep 27 2009In 1970s, Gutman introduced the concept of the energy $\En(G)$ for a simple graph $G$, which is defined as the sum of the absolute values of the eigenvalues of $G$. This graph invariant has attracted much attention, and many lower and upper bounds have ... More

The Laplacian energy of random graphsJun 25 2009Oct 10 2009Gutman {\it et al.} introduced the concepts of energy $\En(G)$ and Laplacian energy $\EnL(G)$ for a simple graph $G$, and furthermore, they proposed a conjecture that for every graph $G$, $\En(G)$ is not more than $\EnL(G)$. Unfortunately, the conjecture ... More

Nordhaus-Gaddum-type theorem for total proper connection number of graphsNov 28 2016Nov 30 2016A graph is said to be \emph{total-colored} if all the edges and the vertices of the graph are colored. A path $P$ in a total-colored graph $G$ is called a \emph{total-proper path} if $(i)$ any two adjacent edges of $P$ are assigned distinct colors; $(ii)$ ... More

3-Rainbow index and forbidden subgraphsOct 18 2016Oct 19 2016A tree in an edge-colored connected graph $G$ is called \emph{a rainbow tree} if no two edges of it are assigned the same color. For a vertex subset $S\subseteq V(G)$, a tree is called an \emph{$S$-tree} if it connects $S$ in $G$. A \emph{$k$-rainbow ... More

Rainbow connection of graphs with diameter 2Jan 14 2011Mar 08 2011A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called a rainbow path if no two edges of the path are colored the same. The rainbow connection number $rc(G)$ of $G$ is the minimum integer $i$ for which there exists ... More

Rainbow connections for planar graphs and line graphsOct 14 2011Nov 14 2011An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in ... More

The (revised) Szeged index and the Wiener index of a nonbipartite graphNov 23 2012Dec 07 2012Hansen et. al. used the computer programm AutoGraphiX to study the differences between the Szeged index $Sz(G)$ and the Wiener index $W(G)$, and between the revised Szeged index $Sz^*(G)$ and the Wiener index for a connected graph $G$. They conjectured ... More

Proper connection number and 2-proper connection number of a graphJul 06 2015Jul 10 2015A path in an edge-colored graph is called a proper path if no two adjacent edges of the path are colored with one same color. An edge-colored graph is called $k$-proper connected if any two vertices of the graph are connected by $k$ internally pairwise ... More

The $k$-proper index of graphsJan 14 2016A tree $T$ in an edge-colored graph is a \emph{proper tree} if any two adjacent edges of $T$ are colored with different colors. Let $G$ be a graph of order $n$ and $k$ be a fixed integer with $2\leq k\leq n$. For a vertex set $S\subseteq V(G)$, a tree ... More

More on rainbow disconnection in graphsOct 23 2018Let $G$ be a nontrivial edge-colored connected graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored the same. An edge-colored graph $G$ is rainbow disconnected if for every two vertices $u$ and $v$, there exists a $u-v$ ... More

Complete solution to a problem on the maximal energy of unicyclic bipartite graphsOct 29 2010The energy of a simple graph $G$, denoted by $E(G)$, is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Denote by $C_n$ the cycle, and $P_n^{6}$ the unicyclic graph obtained by connecting a vertex of $C_6$ with a ... More

Note on the 4- and 5-leaf powersSep 24 2009Motivated by the problem of reconstructing evolutionary history, Nishimura et al. defined $k$-leaf powers as the class of graphs $G=(V,E)$ which has a $k$-leaf root $T$, i.e., $T$ is a tree such that the vertices of $G$ are exactly the leaves of $T$ and ... More

Lower bounds of the skew spectral radii and skew energy of oriented graphsMay 20 2014Jun 12 2014Let $G$ be a graph with maximum degree $\Delta$, and let $G^{\sigma}$ be an oriented graph of $G$ with skew adjacency matrix $S(G^{\sigma})$. The skew spectral radius $\rho_s(G^{\sigma})$ of $G^\sigma$ is defined as the spectral radius of $S(G^\sigma)$. ... More

On oriented graphs with minimal skew energyApr 09 2013Let $S(G^\sigma)$ be the skew-adjacency matrix of an oriented graph $G^\sigma$. The skew energy of $G^\sigma$ is defined as the sum of all singular values of its skew-adjacency matrix $S(G^\sigma)$. In this paper, we first deduce an integral formula for ... More

The matching energy of random graphsDec 22 2014Dec 30 2014The matching energy of a graph was introduced by Gutman and Wagner, which is defined as the sum of the absolute values of the roots of the matching polynomial of the graph. For the random graph $G_{n,p}$ of order $n$ with fixed probability $p\in (0,1)$, ... More

Polynomial reconstruction of the matching polynomialApr 14 2014The matching polynomial of a graph is the generating function of the numbers of its matchings with respect to their cardinality. A graph polynomial is polynomial reconstructible, if its value for a graph can be determined from its values for the vertex-deleted ... More

Characterize graphs with rainbow connection number $m-2$ and $m-3$Dec 11 2013A 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, ... More

Rainbow connection number of dense graphsOct 26 2011An edge-colored graph $G$ is rainbow connected, if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted $rc(G)$, is the smallest number of colors that are needed in ... More

Some extremal results on the colorful monochromatic vertex-connectivity of a graphMar 31 2015A path in a vertex-colored graph is called a \emph{vertex-monochromatic path} if its internal vertices have the same color. A vertex-coloring of a graph is a \emph{monochromatic vertex-connection coloring} (\emph{MVC-coloring} for short), if there is ... More

Constructions of graphs and trees with partially prescribed spectrumNov 07 2016Jan 05 2017It is shown how a connected graph and a tree with partially prescribed spectrum can be constructed. These constructions are based on a recent result of Salez that every totally real algebraic integer is an eigenvalue of a tree. Our result implies that ... More

On graphs with maximum Harary spectral radiusNov 25 2014Let $G$ be a simple graph with vertex set $V(G) = \{v_1 ,v_2 ,\cdots ,v_n\}$. The Harary matrix $RD(G)$ of $G$, which is initially called the reciprocal distance matrix, is an $n \times n$ matrix whose $(i,j)$-entry is equal to $\frac{1}{d_{ij}}$ if $i\not=j$ ... More

New skew Laplacian energy of a simple digraphApr 24 2013For a simple digraph $G$ of order $n$ with vertex set $\{v_1,v_2,\ldots, v_n\}$, let $d_i^+$ and $d_i^-$ denote the out-degree and in-degree of a vertex $v_i$ in $G$, respectively. Let $D^+(G)=diag(d_1^+,d_2^+,\ldots,d_n^+)$ and $D^-(G)=diag(d_1^-,d_2^-,\ldots,d_n^-)$. ... More

Skew Randić Matrix and Skew Randić EnergyJun 05 2014Dec 26 2014Let $G$ 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 ... More

More on the skew-spectra of bipartite graphs and Cartesian products of graphsMay 15 2013Given a graph $G$, let $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. Then the spectrum of $S(G^\sigma)$ is called the skew-spectrum of $G^\sigma$, denoted by $Sp_S(G^\sigma)$. It is known ... More

4-Regular oriented graphs with optimum skew energyApr 03 2013Let $G$ be a simple undirected graph, and $G^\sigma$ be an oriented graph of $G$ with the orientation $\sigma$ and skew-adjacency matrix $S(G^\sigma)$. The skew energy of the oriented graph $G^\sigma$, denoted by $\mathcal{E}_S(G^\sigma)$, is defined ... More

On a Problem of Harary and Schwenk on Graphs with Distinct EigenvaluesMay 23 2014Harary 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 ... More

Graphs with large generalized 3-connectivityJan 14 2012Let $S$ be a nonempty set of vertices of a connected graph $G$. A collection $T_1,..., T_\ell$ of trees in $G$ is said to be internally disjoint trees connecting $S$ if $E(T_i)\cap E(T_j)= \emptyset$ and $V(T_i)\cap V(T_j)=S$ for any pair of distinct ... More

Note on the oriented diameter of graphs with diameter 3Sep 06 2011Sep 27 2011In 1978, Chv\'atal and Thomassen showed that every bridgeless graph with diameter 2 has an orientation with diameter at most 6. They also gave general bounds on the smallest value $f(d)$ such that every bridgeless graph $G$ with diameter $d$ has an orientation ... More

Oriented diameter and rainbow connection number of a graphNov 15 2011Dec 05 2011The oriented diameter of a bridgeless graph $G$ is $\min\{diam(H)\ | H\ is\ an orientation\ of\ G\}$. A path in an edge-colored graph $G$, where adjacent edges may have the same color, is called rainbow if no two edges of the path are colored the same. ... More

Extremality of graph entropy based on degrees of uniform hypergraphs with few edgesSep 27 2017Let $\mathcal{H}$ be a hypergraph with $n$ vertices. Suppose that $d_1,d_2,\ldots,d_n$ are degrees of the vertices of $\mathcal{H}$. The $t$-th graph entropy based on degrees of $\mathcal{H}$ is defined as $$ I_d^t(\mathcal{H}) =-\sum_{i=1}^{n}\left(\frac{d_i^{t}}{\sum_{j=1}^{n}d_j^{t}}\log\frac{d_i^{t}}{\sum_{j=1}^{n}d_j^{t}}\right) ... More

The $(k,\ell)$-proper index of graphsJun 13 2016Jun 17 2016A tree $T$ in an edge-colored graph is called a {\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be an integer with $2\leq k \leq n$. For $S\subseteq V(G)$ and $|S| \ge 2$, an ... More

On two conjectures about the proper connection number of graphsFeb 23 2016Mar 28 2016A path in an edge-colored graph is called proper if no two consecutive edges of the path receive the same color. For a connected graph $G$, the proper connection number $pc(G)$ of $G$ is defined as the minimum number of colors needed to color its edges ... More

On various (strong) rainbow connection numbers of graphsJan 06 2016Apr 20 2017An edge-coloured path is \emph{rainbow} if all the edges have distinct colours. For a connected graph $G$, the \emph{rainbow connection number} $rc(G)$ is the minimum number of colours in an edge-colouring of $G$ such that, any two vertices are connected ... More

The neighbor-scattering number can be computed in polynomial time for interval graphsMar 17 2006Neighbor-scattering number is a useful measure for graph vulnerability. For some special kinds of graphs, explicit formulas are given for this number. However, for general graphs it is shown that to compute this number is NP-complete. In this paper, we ... More

Note on two results on the rainbow connection number of graphsOct 23 2011An edge-colored graph $G$, where adjacent edges may be colored the same, is rainbow connected if any two vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of a connected graph $G$ is the smallest ... More

Monochromatic disconnection of graphsJan 05 2019Jan 26 2019For an edge-colored graph $G$, we call an edge-cut $R$ of $G$ monochromatic if the edges of $R$ are colored with a same color. The graph $G$ is called monochromatically disconnected if any two distinct vertices of $G$ are separated by a monochromatic ... More

Note on the complexity of deciding the rainbow connectedness for bipartite graphsSep 26 2011Sep 27 2011A path in an edge-colored graph is said to be a rainbow path if no two edges on the path have the same color. An edge-colored graph is (strongly) rainbow connected if there exists a rainbow (geodesic) path between every pair of vertices. The (strong) ... More

Monochromatic $k$-edge-connection colorings of graphsOct 28 2018A path in an edge-colored graph $G$ is called monochromatic if any two edges on the path have the same color. For $k\geq 2$, an edge-colored graph $G$ is said to be monochromatic $k$-edge-connected if every two distinct vertices of $G$ are connected by ... More

The asymptotic values of the general Zagreb and Randić indices of trees with bounded maximum degreeApr 11 2010Let $\mathcal {T}^{\Delta}_n$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $\Delta$. Suppose that every tree in $\mathcal {T}^{\Delta}_n$ is equally likely. We show that the number of vertices of ... More

The asymptotic number of occurrences of a subtree in trees with bounded maximum degree and an application to the Estrada indexMay 07 2010Let $\mathcal {T}^{\Delta}_n$ denote the set of trees of order $n$, in which the degree of each vertex is bounded by some integer $\Delta$. Suppose that every tree in $\mathcal {T}^{\Delta}_n$ is equally likely. For any given subtree $H$, we show that ... More

The asymptotic value of Randic index for treesMar 25 2010Let $\mathcal{T}_n$ denote the set of all unrooted and unlabeled trees with $n$ vertices, and $(i,j)$ a double-star. By assuming that every tree of $\mathcal{T}_n$ is equally likely, we show that the limiting distribution of the number of occurrences ... More

Hardness results on generalized connectivityMay 04 2010Let $G$ be a nontrivial connected graph of order $n$ and let $k$ be an integer with $2\leq k\leq n$. For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of edge-disjoint trees $T_1,T_2,...,T_\ell$ in $G$ such that $V(T_i)\cap ... More

Dynamic 3-Coloring of Claw-free GraphsNov 19 2007A {\it dynamic $k$-coloring} of a graph $G$ is a proper $k$-coloring of the vertices of $G$ such that every vertex of degree at least 2 in $G$ will be adjacent to vertices with at least 2 different colors. The smallest number $k$ for which a graph $G$ ... More

Nonexistence of triples of nonisomorphic connected graphs with isomorphic connected $P_3$-graphsNov 23 2007In the paper "Broersma and Hoede, {\it Path graphs}, J. Graph Theory {\bf 13} (1989) 427-444", the authors proposed a problem whether there is a triple of mutually nonisomorphic connected graphs which have an isomorphic connected $P_3$-graph. For a long ... More

NP-completeness of 4-incidence colorability of semi-cubic graphsJul 04 2006The incidence coloring conjecture, proposed by Brualdi and Massey in 1993, states that the incidence coloring number of every graph is at most ${\it \Delta}+2$, where ${\it \Delta}$ is the maximum degree of a graph. The conjecture was shown to be false ... More

Two rainbow connection numbers and the parameter $σ_k(G)$Feb 25 2011Mar 21 2011The rainbow connection number $rc(G)$ and the rainbow vertex-connection number $rvc(G)$ of a graph $G$ were introduced by Chartrand et al. and Krivelevich and Yuster, respectively. Good upper bounds in terms of minimum degree $\delta$ were reported by ... More

The rainbow $k$-connectivity of two classes of graphsJun 22 2009A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of $G$ are colored the same. For a $\kappa$-connected graph $G$ and an integer $k$ with $1\leq k\leq \kappa$, the rainbow $k$-connectivity ... More

Erdös-Gallai-type results for conflict-free connection of graphsDec 27 2018A path in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of its edges. An edge-colored graph is called \emph{conflict-free connected} if there is a conflict-free path between each pair of distinct ... More

Color degree and color neighborhood union conditions for long heterochromatic paths in edge-colored graphsDec 07 2005Let $G$ be an edge-colored graph. A heterochromatic (rainbow, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let $d^c(v)$ denote the color degree and $CN(v)$ denote the color neighborhood of a vertex $v$ of $G$. ... More

On the existence of a rainbow 1-factor in proper coloring of K_{rn}^{(r)}Nov 19 2007El-Zanati et al proved that for any 1-factorization $\mathcal{F}$ of the complete uniform hypergraph $\mathcal {G}=K_{rn}^{(r)}$ with $r\geq 2$ and $n\geq 3$, there is a rainbow 1-factor. We generalize their result and show that in any proper coloring ... More

Color Degree Condition for Large Heterochromatic Matchings in Edge-Colored Bipartite GraphsJun 29 2006Nov 30 2007This paper has been withdrawn by the author(s), due an error in the proof.

Partitioning complete graphs by heterochromatic treesNov 19 2007A {\it heterochromatic tree} is an edge-colored tree in which any two edges have different colors. The {\it heterochromatic tree partition number} of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum positive integer $p$ such that whenever ... More

Sufficient conditions for the existence of perfect heterochromatic matchings in colored graphsNov 24 2005Nov 30 2007This paper has been withdrawn by the author(s), due an error in the proof.

The generalized 3-connectivity of Lexicographic product graphsJul 08 2013Aug 12 2013The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by Chartrand et al., is a natural and nice generalization of the concept of (vertex-)connectivity. In this paper, we prove that for any two connected graphs $G$ and $H$, $\kappa_3(G\circ ... More

A sharp upper bound for the rainbow 2-connection number of 2-connected graphsApr 02 2012Apr 11 2012A path in an edge-colored graph is called {\em rainbow} if no two edges of it are colored the same. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k\leq \ell$, the {\em rainbow $k$-connection number} $rc_k(G)$ of $G$ is defined to be ... More

Rainbow connection number, bridges and radiusMay 04 2011Let $G$ be a connected graph. The notion \emph{the rainbow connection number $rc(G)$} of a graph $G$ was introduced recently by Chartrand et al. Basavaraju et al. showed that for every bridgeless graph $G$ with radius $r$, $rc(G)\leq r(r+2)$, and the ... More

Sharp upper bound for the rainbow connection number of a graph with diameter 2Jun 07 2011Sep 23 2011Let $G$ be a connected graph. The \emph{rainbow connection number $rc(G)$} of a graph $G$ was recently introduced by Chartrand et al. Li et al. proved that for every bridgeless graph $G$ with diameter 2, $rc(G)\leq 5$. They gave examples for which $rc(G)\leq ... More

Sharp upper bound for the rainbow connection numbers of 2-connected graphsMay 21 2011May 26 2011An edge-colored graph $G$, where adjacent edges may be colored the same, is rainbow connected if any two vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of a connected graph $G$ is the smallest ... More

On strong rainbow connection numberOct 29 2010A 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. For any two vertices $u$ and $v$ of $G$, a rainbow $u-v$ geodesic in $G$ is a rainbow $u-v$ path of length $d(u,v)$, ... More

Note on the Rainbow $k$-Connectivity of Regular Complete Bipartite GraphsApr 14 2010A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of the path are colored the same. For a $\kappa$-connected graph $G$ and an integer $k$ with $1\leq k\leq \kappa$, the rainbow ... More

Upper bounds for the rainbow connection numbers of line graphsJan 02 2010A path in an edge-colored graph $G$, where adjacent edges may be colored the same, is called a rainbow path if no two edges of it are colored the same. A nontrivial connected graph $G$ is rainbow connected if for any two vertices of $G$ there is a rainbow ... More

A Turán-type problem on degree sequenceFeb 07 2013Given $p\geq 0$ and a graph $G$ whose degree sequence is $d_1,d_2,\ldots,d_n$, let $e_p(G)=\sum_{i=1}^n d_i^p$. Caro and Yuster introduced a Tur\'an-type problem for $e_p(G)$: given $p\geq 0$, how large can $e_p(G)$ be if $G$ has no subgraph of a particular ... More

The minimal size of a graph with given generalized 3-edge-connectivityJan 18 2012Jul 09 2013For $S\subseteq V(G)$ and $|S|\geq 2$, $\lambda(S)$ is the maximum number of edge-disjoint trees connecting $S$ in $G$. For an integer $k$ with $2\leq k\leq n$, the \emph{generalized $k$-edge-connectivity} $\lambda_k(G)$ of $G$ is then defined as $\lambda_k(G)= ... More

On extremal graphs with at most $\ell$ internally disjoint Steiner trees connecting any n-1 verticesApr 13 2013The concept of maximum local connectivity $\bar {\kappa}$ of a graph was introduced by Bollob\'{a}s. One of the problems about it is to determine the largest number of edges $f(n;\bar{\kappa}\leq \ell)$ for graphs of order $n$ that have local connectivity ... More

On extremal graphs with exactly one Steiner tree connecting any $k$ verticesJan 20 2013The problem of determining the largest number $f(n;\bar{\kappa}\leq \ell)$ of edges for graphs with $n$ vertices and maximal local connectivity at most $\ell$ was considered by Bollob\'{a}s. Li et al. studied the largest number $f(n;\bar{\kappa}_3\leq2)$ ... More

Conflict-free connection number of random graphsSep 10 2018An edge-colored graph $G$ is conflict-free connected if any two of its vertices are connected by a path which contains a color used on exactly one of its edges. The conflict-free connection number of a connected graph $G$, denoted by $cfc(G)$, is the ... More

Long heterochromatic paths in heterochromatic triangle free graphsApr 29 2008In this paper, graphs under consideration are always edge-colored. We consider long heterochromatic paths in heterochromatic triangle free graphs. Two kinds of such graphs are considered, one is complete graphs with Gallai colorings, i.e., heterochromatic ... More

On a question on graphs with rainbow connection number 2Sep 23 2011Sep 26 2011For a connected graph $G$, the \emph{rainbow connection number $rc(G)$} of a graph $G$ was introduced by Chartrand et al. In "Chakraborty et al., Hardness and algorithms for rainbow connection, J. Combin. Optim. 21(2011), 330--347", Chakraborty et al. ... More

Rainbow vertex-connection number of 2-connected graphsOct 26 2011The {\em rainbow vertex-connection number}, $rvc(G)$, of a connected graph $G$ 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. ... More

Upper bounds involving parameter $σ_2$ for the rainbow connectionJan 17 2011For a graph $G$, we define $\sigma_2(G)=min \{d(u)+d(v)| u,v\in V(G), uv\not\in E(G)\}$, or simply denoted by $\sigma_2$. A edge-colored graph is rainbow edge-connected if any two vertices are connected by a path whose edges have distinct colors, which ... More

A solution to a conjecture on the rainbow connection numberDec 13 2010Dec 14 2010For a graph $G$, Chartrand et al. defined the rainbow connection number $rc(G)$ and the strong rainbow connection number $src(G)$ in "G. Charand, G.L. John, K.A. Mckeon, P. Zhang, Rainbow connection in graphs, Mathematica Bohemica, 133(1)(2008) 85-98". ... More

Strong conflict-free connection of graphsJan 24 2019A path $P$ in an edge-colored graph is called \emph{a conflict-free path} if there exists a color used on only one of the edges of $P$. An edge-colored graph $G$ is called \emph{conflict-free connected} if for each pair of distinct vertices of $G$ there ... More

Graphs with large generalized (edge-)connectivityMay 06 2013Aug 31 2015The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$, introduced by Hager in 1985, is a nice generalization of the classical connectivity. Recently, as a natural counterpart, we proposed the concept of generalized $k$-edge-connectivity $\lambda_k(G)$. ... More

Nordhaus-Gaddum-type results for the generalized edge-connectivity of graphsDec 30 2012Let $G$ be a graph, $S$ be a set of vertices of $G$, and $\lambda(S)$ be the maximum number $\ell$ of pairwise edge-disjoint trees $T_1, T_2,..., T_{\ell}$ in $G$ such that $S\subseteq V(T_i)$ for every $1\leq i\leq \ell$. The generalized $k$-edge-connectivity ... More

Bicyclic graphs with maximal revised Szeged indexApr 12 2011The revised Szeged index $Sz^*(G)$ is defined as $Sz^*(G)=\sum_{e=uv \in E}(n_u(e)+ n_0(e)/2)(n_v(e)+ n_0(e)/2),$ where $n_u(e)$ and $n_v(e)$ are, respectively, the number of vertices of $G$ lying closer to vertex $u$ than to vertex $v$ and the number ... More

Note for Nikiforov's two conjectures on the energy of treesJun 04 2009The energy $E$ of a graph is defined to be the sum of the absolute values of its eigenvalues. Nikiforov in {\it ``V. Nikiforov, The energy of $C_4$-free graphs of bounded degree, Lin. Algebra Appl. 428(2008), 2569--2573"} proposed two conjectures concerning ... More

Randić index, diameter and the average distanceJun 29 2009The Randi\'c index of a graph $G$, denoted by $R(G)$, is defined as the sum of $1/\sqrt{d(u)d(v)}$ over all edges $uv$ of $G$, where $d(u)$ denotes the degree of a vertex $u$ in $G$. In this paper, we partially solve two conjectures on the Randi\'c index ... More

Note on rainbow connection number of dense graphsOct 05 2011An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in ... More

Rainbow connections of graphs -- A surveyJan 30 2011Feb 01 2011The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers that dealt ... More

Rainbow connection numbers of complementary graphsNov 20 2010Dec 23 2010A 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, ... More

Rainbow connection in $3$-connected graphsOct 29 2010An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number of a connected graph $G$, denoted by $rc(G)$, is the smallest number of colors that are needed in ... More

Long rainbow path in properly edge-colored complete graphsMar 16 2015Let $G$ be an edge-colored graph. A rainbow (heterochromatic, or multicolored) path of $G$ is such a path in which no two edges have the same color. Let the color degree of a vertex $v$ be the number of different colors that are used on the edges incident ... More

A survey of recent results in (generalized) graph entropiesMay 18 2015May 19 2015The entropy of a graph was first introduced by Rashevsky \cite{Rashevsky} and Trucco \cite{Trucco} to interpret as the structural information content of the graph and serve as a complexity measure. In this paper, we first state a number of definitions ... More

A proof of the conjecture on hypoenergetic graphs with maximum degree $Δ\leq 3$Jun 15 2009Jun 16 2009The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. A graph $G$ of order $n$ is said to be hypoenergetic if $E(G)<n$. Majstorovi\'{c} et al. conjectured that complete bipartite graph $K_{2,3}$ is the only ... More

Hypoenergetic and strongly hypoenergetic treesMay 25 2009May 26 2009The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of the eigenvalues of $G$. An $n$-vertex graph is said to be hypoenergetic if $E(G)<n$ and strongly hypoenergetic if $E(G)<n-1$. In this paper, we consider hypoenergetic and ... More

The (vertex-)monochromatic index of a graphMar 17 2016Mar 19 2016A tree $T$ in an edge-colored graph $H$ is called a \emph{monochromatic tree} if all the edges of $T$ have the same color. For $S\subseteq V(H)$, a \emph{monochromatic $S$-tree} in $H$ is a monochromatic tree of $H$ containing the vertices of $S$. For ... More

Hardness results for rainbow disconnection of graphsNov 21 2018Dec 04 2018Let $G$ be a nontrivial connected, edge-colored graph. An edge-cut $S$ of $G$ is called a rainbow cut if no two edges in $S$ are colored with a same color. An edge-coloring of $G$ is a rainbow disconnection coloring if for every two distinct vertices ... More

Monochromatic and heterochromatic subgraph problems in a randomly colored graphNov 24 2007Let $K_n$ be the complete graph with $n$ vertices and $c_1, c_2, ..., c_r$ be $r$ different colors. Suppose we randomly and uniformly color the edges of $K_n$ in $c_1, c_2, ..., c_r$. Then we get a random graph, denoted by $\mathcal{K}_n^r$. In the paper, ... More

On the rainbow vertex-connectionDec 16 2010A vertex-colored graph is {\it rainbow vertex-connected} if any two vertices are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection} of a connected graph ... More

Partitioning 3-edge-colored complete equi-bipartite graphs by monochromatic trees under a color degree conditionJan 01 2008Jan 03 2008The monochromatic tree partition number of an $r$-edge-colored graph $G$, denoted by $t_r(G)$, is the minimum integer $k$ such that whenever the edges of $G$ are colored with $r$ colors, the vertices of $G$ can be covered by at most $k$ vertex-disjoint ... More

Rainbow number of matchings in regular bipartite graphsNov 19 2007Given a graph $G$ and a subgraph $H$ of $G$, let $rb(G,H)$ be the minimum number $r$ for which any edge-coloring of $G$ with $r$ colors has a rainbow subgraph $H$. The number $rb(G,H)$ is called the rainbow number of $H$ with respect to $G$. Denote $mK_2$ ... More