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Monochromatic disconnection: Erdős-Gallai-type problems and product graphsApr 18 2019For an edge-colored graph $G$, we call an edge-cut $M$ of $G$ monochromatic if the edges of $M$ 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

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

All Connected Graphs with Maximum Degree at Most 3 whose Energies are Equal to the Number of VerticesJul 08 2009The energy $E(G)$ of a graph $G$ is defined as the sum of the absolute values of its eigenvalues. Let $S_2$ be the star of order 2 (or $K_2$) and $Q$ be the graph obtained from $S_2$ by attaching two pendent edges to each of the end vertices of $S_2$. ... 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

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

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

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

The asymptotic value of graph energy for random graphs with degree-based weightsApr 30 2019May 14 2019In this paper, we investigate the energy of weighted random graphs $G_{n,p}(f)$, in which each edge $ij$ takes the weight $f(d_i,d_j)$, where $d_v$ is a random variable, the degree of vertex $v$ in the random graph $G_{n,p}$ of Erd\"{o}s--R\'{e}nyi model, ... More

Sharp bounds for the generalized connectivity $κ_3(G)$Jun 17 2009Let $G$ 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

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

Hardness result for the total rainbow $k$-connection of graphsNov 19 2015A path in a total-colored graph is called \emph{total rainbow} if its edges and internal vertices have distinct colors. For an $\ell$-connected graph $G$ and an integer $k$ with $1\leq k \leq\ell$, the \emph{total rainbow $k$-connection number} of $G$, ... More

The $k$-proper index of complete bipartite and complete multipartite graphsJul 30 2016Dec 06 2016Let $G$ be a nontrivial connected graph of order $n$ with an edge-coloring $c:E(G)\rightarrow\{1,2,\dots,t\}$,$t\in\mathbb{N}$, where adjacent edges may be colored with the same color. A tree $T$ in $G$ is a \emph{proper tree} if no two adjacent edges ... More

Rainbow vertex-connection and forbidden subgraphsFeb 02 2016A path in a vertex-colored graph is called \emph{vertex-rainbow} if its internal vertices have pairwise distinct colors. A graph $G$ is \emph{rainbow vertex-connected} if for any two distinct vertices of $G$, there is a vertex-rainbow path connecting ... 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

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

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

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

The generalized 3-edge-connectivity of lexicographic product graphsJan 10 2014The generalized $k$-edge-connectivity $\lambda_k(G)$ of a graph $G$ is a generalization of the concept of edge-connectivity. The lexicographic product of two graphs $G$ and $H$, denoted by $G\circ H$, is an important graph product. In this paper, we mainly ... More

On maximum Estrada indices of bipartite graphs with some given parametersApr 22 2014The Estrada index of a graph $G$ is defined as $EE(G)=\sum_{i=1}^ne^{\lambda_i}$, where $\lambda_1,$ $ \lambda_2,\ldots, \lambda_n$ are the eigenvalues of the adjacency matrix of $G$. In this paper, we characterize the unique bipartite graph with maximum ... More

Proper connection number and connected dominating setsJan 23 2015The proper connection number $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors needed to color its edges, so that every pair of distinct vertices of $G$ is connected by at least one path in $G$ such that no two adjacent edges ... More

Further hardness results on the rainbow vertex-connection number of graphsOct 10 2011A vertex-colored graph $G$ is {\it rainbow vertex-connected} if any pair of vertices in $G$ are connected by a path whose internal vertices have distinct colors, which was introduced by Krivelevich and Yuster. The {\it rainbow vertex-connection number} ... More

The $(k,\ell)$-rainbow index of random graphsOct 10 2013A tree in an edge colored graph is said to be a rainbow tree if no two edges on the tree share the same color. Given two positive integers $k$, $\ell$ with $k\geq 3$, the \emph{$(k,\ell)$-rainbow index} $rx_{k,\ell}(G)$ of $G$ is the minimum number of ... More

The generalized 3-connectivity of random graphsMar 21 2013The generalized connectivity of a graph $G$ was introduced by Chartrand et al. Let $S$ be a nonempty set of vertices of $G$, and $\kappa(S)$ be defined as the largest number of internally disjoint trees $T_1, T_2, \cdots, T_k$ connecting $S$ in $G$. Then ... More

More on total monochromatic connection of graphsApr 08 2016A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are ... More

Good upper bounds for the total rainbow connection of graphsJan 08 2015A total-colored graph is a graph $G$ such that both all edges and all vertices of $G$ are colored. A path in a total-colored graph $G$ is a total rainbow path if its edges and internal vertices have distinct colors. A total-colored graph $G$ is total-rainbow ... More

The 3-rainbow index and connected dominating setsApr 09 2014Apr 14 2014A tree in an edge-colored graph is said to be rainbow if no two edges on the tree share the same color. An edge-coloring of $G$ is called 3-rainbow if for any three vertices in $G$, there exists a rainbow tree connecting them. The 3-rainbow index $rx_3(G)$ ... More

Erdős-Gallai-type results for colorful monochromatic connectivity of a graphDec 25 2014A 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 ... More

Erdős-Gallai-type results for total monochromatic connection of graphsDec 16 2016A graph is said to be {\it total-colored} if all the edges and the vertices of the graph are colored. A total-coloring of a graph is a {\it total monochromatically-connecting coloring} ({\it TMC-coloring}, for short) if any two vertices of the graph are ... More

Conflict-free connection numbers of line graphsMay 15 2017A path in an edge-colored graph is called \emph{conflict-free} if it contains at least one color used on exactly one of its edges. An edge-colored graph $G$ is \emph{conflict-free connected} if for any two distinct vertices of $G$, there is a conflict-free ... More

On (strong) proper vertex-connection of graphsMay 19 2015A path in a vertex-colored graph is a {\it vertex-proper path} if any two internal adjacent vertices differ in color. A vertex-colored graph is {\it proper vertex $k$-connected} if any two vertices of the graph are connected by $k$ disjoint vertex-proper ... 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

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

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

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

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

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

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

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

Erdős-Gallai-type results for the rainbow disconnection number of graphsJan 08 2019Let $G$ be a nontrivial connected and edge-colored graph. An edge-cut $R$ of $G$ is called a rainbow cut if no two edges of it are colored with a same color. An edge-colored graph $G$ is called rainbow disconnected if for every two distinct vertices $u$ ... 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

Rainbow connection number and independence number of a graphApr 19 2012Apr 03 2013Let $G$ be an edge-colored connected graph. A path of $G$ is called rainbow if its every edge is colored by a distinct color. $G$ is called rainbow connected if there exists a rainbow path between every two vertices of $G$. The minimum number of colors ... 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

Rainbow connection number and the number of blocksNov 01 2012Nov 05 2012An edge-colored graph $G$ is rainbow connected if every pair of vertices of $G$ are connected by a path whose edges have distinct colors. The rainbow connection number $rc(G)$ of $G$ is defined to be the minimum integer $t$ such that there exists an edge-coloring ... 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

Note on a relation between Randic index and algebraic connectivityDec 22 2010A conjecture of AutoGraphiX on the relation between the Randi\'c index $R$ and the algebraic connectivity $a$ of a connected graph $G$ is: $$\frac R a\leq (\frac{n-3+2\sqrt{2}}{2})/(2(1- \cos {\frac{\pi}{n}})) $$ with equality if and only if $G$ is $P_n$, ... More

A survey on the generalized connectivity of graphsJul 08 2012Aug 31 2015The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity $\lambda_k(G)$, recently. In this paper we summarize the known ... 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

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

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

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

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

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

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.

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 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

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

Conflict-free (vertex)-connection numbers of graphs with small diametersFeb 28 2019Apr 10 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

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

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

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

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

A Source-level Energy Optimization Framework for Mobile ApplicationsAug 18 2016Energy efficiency can have a significant influence on user experience of mobile devices such as smartphones and tablets. Although energy is consumed by hardware, software optimization plays an important role in saving energy, and thus software developers ... More