Let $G$ be a graph with $n$ vertices. We denote the largest signless Laplacian eigenvalue of $G$ by $q_1(G)$ and Laplacian eigenvalues of $G$ by $μ_1(G)\ge\cdots\geμ_{n-1}(G)\geμ_n(G)=0$. It is a conjecture on Laplacian spread of graphs that $μ_1(G)-μ_{n-1}(G)\le n-1$ or equivalently $μ_1(G)+μ_1(\Gb)\le2n-1$. We prove the conjecture for bipartite graphs. Also we show that for any bipartite graph $G$, $μ_1(G)μ_1(\Gb)\le n(n-1)$. Aouchiche and Hansen [A survey of Nordhaus--Gaddum type relations, Discrete Appl. Math. 161 (2013), 466--546] conjectured that %for any graph $G$ with $n$ vertices, $q_1(G)+q_1(\Gb)\le3n-4$ and $q_1(G)q_1(\Gb)\le2n(n-2)$. We prove the former and disprove the latter by constructing a family of graphs $H_n$ where $q_1(H_n)q_1(\ov{H_n})$ is about $2.15n^2+O(n)$.
In 1970 Nosal gave upper and lower bounds on the sum of the spectral radii of a graph and its complement. We generalize this problem to other eigenvalues and give a number of bounds. We essentially solve the corresponding problem for the second eigenvalues.
Terpai [22] proved the Nordhaus-Gaddum bound that $μ(G) + μ(\overline{G}) \le 4n/3 - 1$, where $μ(G)$ is the spectral radius of a graph $G$ with $n$ vertices. Let $s^+$ denote the sum of the squares of the positive eigenvalues of $G$. We prove that $\sqrt{s^{+}(G)} + \sqrt{s^+(\overline{G})} < \sqrt{2}n$ and conjecture that $\sqrt{s^{+}(G)} + \sqrt{s^+(\overline{G})} \le 4n/3 - 1.$ We have used AutoGraphiX and Wolfram Mathematica to search for a counter-example. We also consider Nordhaus-Gaddum bounds for $s^+$ and bounds for the Randić index.
A traditional Nordhaus-Gaddum problem for a graph parameter $β$ is to find a (tight) upper or lower bound on the sum or product of $β(G)$ and $β(\bar{G})$ (where $\bar{G}$ denotes the complement of $G$). An $r$-decomposition $G_1,\dots,G_r$ of the complete graph $K_n$ is a partition of the edges of $K_n$ among $r$ spanning subgraphs $G_1,\dots,G_r$. A traditional Nordhaus-Gaddum problem can be viewed as the special case for $r=2$ of a more general $r$-part sum or product Nordhaus-Gaddum type problem. We determine the values of the $r$-part sum and product upper bounds asymptotically as $n$ goe…
The upper and lower Nordhaus-Gaddum bounds over all graphs for the power domination number follow from known bounds on the domination number and examples. In this note we improve the upper sum bound for the power domination number substantially for graphs having the property that both the graph and its complement must be connected. For these graphs, our bound is tight and is also significantly better than the corresponding bound for domination number. We also improve the product upper bound for the power domination number for graphs with certain properties.
We study the Nordhaus-Gaddum type results for $(k-1,k,j)$ and $k$-domination numbers of a graph $G$ and investigate these bounds for the $k$-limited packing and $k$-total limited packing numbers in graphs. As the special case $(k-1,k,j)=(1,2,0)$ we give an upper bound on $dd(G)+dd(\overline{G})$ stronger than that presented by Harary and Haynes (1996). Moreover, we establish upper bounds on the sum and product of packing and open packing numbers and characterize all graphs attaining these bounds.
A dominating set S of graph G is called metric-locating-dominating if it is also locating, that is, if every vertex v is uniquely determined by its vector of distances to the vertices in S. If moreover, every vertex v not in S is also uniquely determined by the set of neighbors of v belonging to S, then it is said to be locating-dominating. Locating, metric-locating-dominating and locating-dominating sets of minimum cardinality are called b-codes, e-codes and l-codes, respectively. A Nordhaus-Gaddum bound is a tight lower or upper bound on the sum or product of a parameter of a graph G and its…
A vertex-colored graph $G$ is rainbow vertex-connected if any pair of distinct vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection number of $G$, denoted by $rvc(G)$, is the minimum number of colors that are needed to make $G$ rainbow vertex-connected. In this paper we give a Nordhaus-Gaddum-type result of the rainbow vertex-connection number. We prove that when $G$ and $\bar{G}$ are both connected, then $2\leq rvc(G)+rvc(\bar{G})\leq n-1$. Examples are given to show that both the upper bound and the lower bound are best possible for all…
California State University Long Beach（加州州立大学长滩分校）；California State University, San Bernardino（加州州立大学圣贝纳迪诺分校）；Fort Lewis College（福特路易斯学院）；American Institute of Mathematics（美国数学研究所）；Iowa State University（爱荷华州立大学）；Purdue University（普渡大学）；Kutztown University of Pennsylvania（宾夕法尼亚库兹敦大学）
An 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 $G$, denoted $rc(G)$, is the minimum number of colors that are used to make $G$ rainbow connected. In this paper we give a Nordhaus-Gaddum-type result for the rainbow connection number. We prove that if $G$ and $\bar{G}$ are both connected, then $4\leq rc(G)+rc(\bar{G})\leq n+2$. Examples are given to show that the upper bound is sharp for all $n\geq 4$, and the lower bound is sharp for all $n\geq 8$. For the rest small $n=4,5,6,7,$ we al…
Let $G$ be a graph of order $n$ and let $μ_{1}\left(G\right) \geq \cdots\geqμ_{n}\left(G\right) $ be the eigenvalues of its adjacency matrix. This note studies eigenvalue problems of Nordhaus-Gaddum type. Let $\overline{G}$ be the complement of a graph $G.$ It is shown that if $s\geq2$ and $n\geq15\left(s-1\right) ,$ then \[ \left\vert μ_{s}\left(G\right) \right\vert +|μ_{s}(\overline{G})|\,\leq n/\sqrt{2\left(s-1\right)}-1. \] Also if $s\geq1$ and $n\geq4^{s},$ then \[ \left\vert μ_{n-s+1}\left(G\right) \right\vert +|μ_{n-s+1}(\overline {G})|\,\leq n/\sqrt{2s}+1. \] If $s=2^{k}+1$ for some in…
Let $G$ be a graph, $S$ be a set of vertices of $G$, and $λ(S)$ 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 $λ_k(G)$ of $G$ is defined as $λ_k(G)= min\{λ(S) | S\subseteq V(G) \ and \ |S|=k\}$. Thus $λ_2(G)=λ(G)$. In this paper, we consider the Nordhaus-Gaddum-type results for the parameter $λ_k(G)$. We determine sharp upper and lower bounds of $λ_k(G)+λ_k(\bar{G})$ and $λ_k(G)... λ_k(\bar{G})$ for a graph $G$ of order $n$, as well as for a graph of …
Let $H=(V(H),E(H))$ be a graph. A $k$-coloring of $H$ is a mapping $π: V(H) \longrightarrow \{1,2,\ldots, k\}$, if each color class induces a $K_2$-free subgraph. For a graph $G$ of order at least $2$, a $G$-free $k$-coloring of $H$, is a mapping $π: V(H) \longrightarrow \{1,2,\ldots,k\}$, so that the induced subgraph by each color class of $π$, contains no copy of $G$. The $G$-free chromatic number of $H$, is the minimum number $k$, so that it has a $G$-free $k$-coloring, and denoted by $χ_G(H)$. In this paper, we give some bounds and attributes on the $G$-free chromatic number of graphs, in …
We prove that for every graph $G$ with $n$ vertices, the treewidth of $G$ plus the treewidth of the complement of $G$ is at least $n-2$. This bound is tight.
A dominating set in a graph $G$ is a set $S$ of vertices such that every vertex of $G$ is either in $S$ or is adjacent to a vertex in $S$. Nordhaus-Gaddum inequailties relate a graph $G$ to its complement $\bar{G}$. In this spirit Wagner proved that any graph $G$ on $n$ vertices satisfies $\partial(G)+\partial(\bar{G})\geq 2^n$ where $\partial(G)$ is the number of dominating sets in a graph $G$. In the same paper he comments that an upper bound for $\partial(G)+\partial(\bar{G})$ among all graphs on $n$ vertices seems to be much more difficult. Here we prove an upper bound on $\partial(G)+\par…
We study the number of lines in hypergraphs in a more symmetric setting, where both the hypergraph and its complement are considered. In the general case and in some special cases, the lower bounds on the number of lines are much higher than their counterparts in single hypergraph setting or admit more elegant proofs. We show that the minimum value of product of the number of lines in both hypergraphs on $n$ points is easily determined as $\binom{n}{2}$; and the minimum value of their sum is between $Ω(n)$ and $O(n \log n)$. We also study some restricted classes of hypergraphs; and determine t…
Let $G$ be a graph of order $n$, and let $q_1(G)\geq q_2(G)\geq\cdots\geq q_n(G)$ denote the signless Laplacian eigenvalues of $G$. Ashraf and Tayfeh-Rezaie [Electron. J. Combin. 21 (3) (2014) \#P3.6] showed that $q_1(G)+q_1(\overline{G})\leq 3n-4$, with equality holding if and only if $G$ or $\overline{G}$ is the star $K_{1,n-1}$. In this paper, we discuss the following problem: for $n\geq6$, does $q_2(G)+q_2(\overline{G})\leq 2n-5$ always hold? We provide positive answers to this problem for the graphs with disconnected complements and the bipartite graphs, and determine the graphs attaining…
We propose a Nordhaus-Gaddum conjecture for $q(G)$, the minimum number of distinct eigenvalues of a symmetric matrix corresponding to a graph $G$: for every graph $G$ excluding four exceptions, we conjecture that $q(G)+q(G^c)\le |G|+2$, where $G^c$ is the complement of $G$. We compute $q(G^c)$ for all trees and all graphs $G$ with $q(G)=|G|-1$, and hence we verify the conjecture for trees, unicyclic graphs, graphs with $q(G)\le 4$, and for graphs with $|G|\le 7$.
A graph is said to be \emph{total-colored} if all the edges and the vertices of the graph are colored. A total-colored graph is \emph{total-rainbow connected} if any two vertices of the graph are connected by a path whose edges and internal vertices have distinct colors. For a connected graph $G$, the \emph{total-rainbow connection number} of $G$, denoted by $trc(G)$, is the minimum number of colors required in a total-coloring of $G$ to make $G$ total-rainbow connected. In this paper, we first characterize the graphs having large total-rainbow connection numbers. Based on this, we obtain a No…
A graph G is an NG-graph if χ(G) + χ(G complement) = |V(G)| + 1. We characterize NG-graphs solely from degree sequences leading to a linear-time recognition algorithm. We also explore the connections between NG-graphs and split graphs. There are three types of NG-graphs and split graphs can also be divided naturally into two categories, balanced and unbalanced. We characterize each of these five classes by degree sequence. We construct bijections between classes of NG-graphs and balanced and unbalanced split graphs which, together with the known formula for the number of split graphs on n vert…
In 1956, Nordhaus and Gaddum gave lower and upper bounds on the sum and the product of the chromatic number of a graph and its complement, in terms of the order of the graph. Since then, any bound on the sum and/or the product of an invariant in a graph $G$ and the same invariant in the complement $G^c$ of $G$ is called a Nordhaus-Gaddum type inequality or relation. The Nordhaus-Gaddum type inequalities for connectivity have been studied by several authors. For a bipartite graph $G=G[X,Y]$ with bipartition ($X,Y$), its bipartite complementary graph $G^{bc}$ is a bipartite graph with $V(G^{bc})…
An edge-colored graph $G$ is \emph{conflict-free connected} if, between each pair of distinct vertices, there exists a path containing a color used on exactly one of its edges. The \emph{conflict-free connection number} of a connected graph $G$, denoted by $cfc(G)$, is defined as the smallest number of colors that are needed in order to make $G$ conflict-free connected. In this paper, we determine all trees $T$ of order $n$ for which $cfc(T)=n-t$, where $t\geq 1$ and $n\geq 2t+2 $. Then we prove that $1\leq cfc(G)\leq n-1$ for a connected graph $G$, and characterize the graphs $G$ with $cfc(G)…
A 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; $(ii)$ any two adjacent internal vertices of $P$ are assigned distinct colors; $(iii)$ any internal vertex of $P$ is assigned a distinct color from its incident edges of $P$. The total-colored graph $G$ is \emph{total-proper connected} if any two distinct vertices of $G$ are connected by a total-proper path. The \emph{total-proper connection number}…
Nordhaus and Gaddum proved, for any graph G, that the chromatic number of G plus the chromatic number of G complement is less than or equal to the number of vertices in G plus 1. Finck characterized the class of graphs that satisfy equality in this bound. In this paper, we provide a new characterization of this class of graphs, based on vertex degrees, which yields a new polynomial-time recognition algorithm and efficient computation of the chromatic number of graphs in this class. Our motivation comes from our theorem that generalizes the Nordhaus-Gaddum theorem to the distinguishing chromati…
On the Burning Game: Nordhaus-Gaddum Bounds and Graph Products
University of Primorska（波雷克大学）；Faculty of Mathematics and Physics, University Ljubljana（卢布尔雅那大学数学物理学院）；Institute of Mathematics, Physics and Mechanics, Ljubljana（卢布尔雅那数学、物理与力学研究所）；Faculty of Natural Sciences and Mathematics, University of Maribor（马里博尔大学自然科学与数学学院）；University of Rhode Island（罗德岛大学）；Department of Mathematics and Informatics, Faculty of Sciences, University of Novi Sad（诺维萨德大学理学院数学与信息系）
Let $η(G)$ be the number of connected induced subgraphs in a graph $G$, and $\overline{G}$ the complement of $G$. We prove that $η(G)+η(\overline{G})$ is minimum, among all $n$-vertex graphs, if and only if $G$ has no induced path on four vertices. Since the $n$-vertex star $S_n$ with maximum degree $n-1$ is the unique tree of diameter $2$, $η(S_n)+η(\overline{S_n})$ is minimum among all $n$-vertex trees, while the maximum is shown to be achieved only by the tree whose degree sequence is $(\lceil n/2\rceil,\lfloor n/2\rfloor,1,\dots,1)$. Furthermore, we prove that every graph $G$ of order $n\g…
The fractional matching number of a graph G, is the maximum size of a fractional matching of G. The following sharp lower bounds for a graph G of order n are proved, and all extremal graphs are characterized in this paper. (1)The sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than n/2 , where n is not less than 2. (2) If G and its complement are non-empty, then the sum of the fractional matching number of a graph G and the fractional matching number of its complement is not less than (n+1)/2, where n is not less than 28. (3) …
We extend the Grundy number and the ochromatic number, parameters on graph colorings, to digraph colorings, we call them {\emph{digrundy number}} and {\emph{diochromatic number}}, respectively. First, we prove that for every digraph the diochromatic number equals the digrundy number (as it happen for graphs). Then, we prove the interpolation property and the Nordhaus-Gaddum relations for the digrundy number, and improve the Nordhaus-Gaddum relations for the dichromatic and diachromatic numbers bounded previously by the authors in [Electron. J. Combin. 25 (2018) no. 3, Paper {\#} 3.51, 17 pp.]
Building upon the notion of Gutman index $\operatorname{SGut}(G)$, Mao and Das recently introduced the Steiner Gutman index by incorporating Steiner distance for a connected graph $G$. The \emph{Steiner Gutman $k$-index} $\operatorname{SGut}_k(G)$ of $G$ is defined by $\operatorname{SGut}_k(G)$ $=\sum_{S\subseteq V(G), \ |S|=k}\left(\prod_{v\in S}deg_G(v)\right) d_G(S)$, in which $d_G(S)$ is the Steiner distance of $S$ and $deg_G(v)$ is the degree of $v$ in $G$. In this paper, we derive new sharp upper and lower bounds on $\operatorname{SGut}_k$, and then investigate the Nordhaus-Gaddum-type r…
We introduce a characterization for split graphs by using edge contraction. Then, we use it to prove that any ($2K_{2}$, claw)-free graph with $α(G) \geq 3$ is a split graph. Also, we apply it to characterize any pseudo-split graph. Finally, by using edge contraction again, we characterize unbalanced split graphs which we use to characterize the Nordhaus-Gaddum graphs.
A Nordhaus-Gaddum-type result is a (tight) lower or upper bound on the sum or product of a parameter of a graph and its complement. In this paper some variations are considered. First, recall their theorem, which gives bounds on the sum and the product of the chromatic number of a graph with that of its complement. In this paper, we provide a new characterization of the other class of graphs.
Let $λ_1,λ_2,\cdots,λ_n$ be the eigenvalues of the distance matrix of a connected graph $G$. The distance Estrada index of $G$ is defined as $DEE(G)=\sum_{i=1}^ne^{λ_i}$. In this note, we present new lower and upper bounds for $DEE(G)$. In addition, a Nordhaus-Gaddum type inequality for $DEE(G)$ is given.
Let $(n^+, n^0, n^-)$ denote the inertia of a graph $G$ with $n$ vertices. Nordhaus-Gaddum bounds are known for inertia, except for an upper bound for $n^-$. We conjecture that for any graph \[ n^-(G) + n^-(\bar{G}) \le 1.5(n - 1), \] and prove this bound for various classes of graphs and for almost all graphs. We consider the relationship between this bound and the number of eigenvalues that lie within the interval $-1$ to $0$, which we denote $n_{(-1,0)}(G)$. We conjecture that for any graph \[ n_{(-1,0)}(G) \le 0.5(n - 1). \] and prove this bound for almost all graphs. We also investigate e…
A set $B$ of vertices in a graph $G$ is called a \emph{$k$-limited packing} if for each vertex $v$ of $G$, its closed neighbourhood has at most $k$ vertices in $B$. The \emph{$k$-limited packing number} of a graph $G$, denoted by $L_k(G)$, is the largest number of vertices in a $k$-limited packing in $G$. The concept of the $k$-limited packing of a graph was introduced by Gallant et al., which is a generalization of the well-known packing of a graph. In this paper, we present some tight bounds for the $k$-limited packing number of a graph in terms of its order, diameter, girth, and maximum deg…
In 1993 Hong asked what are the best bounds on the $k$'th largest eigenvalue $λ_{k}(G)$ of a graph $G$ of order $n$. This challenging question has never been tackled for any $2<k<n$. In the present paper tight bounds are obtained for all $k>2,$ and even tighter bounds are obtained for the $k$'th largest singular value $λ_{k}^{\ast}(G).$ Some of these bounds are based on Taylor's strongly regular graphs, and other on a method of Kharaghani for constructing Hadamard matrices. The same kind of constructions are applied to other open problems, like Nordhaus-Gaddum problems of the kind: How large c…
We shall study degree-monotone paths in graphs, a problem inspired by the celebrated theorem of Erd{ő}s-Szekeres concerning the longest monotone subsequence of a given sequence of numbers. A path P in a graph G is said to be a degree monotone path if the sequence of degrees of the vertices in P in the order they appear in P is monotonic. In this paper we shall consider these three problem related to this parameter: 1. Find bounds on $mp(G)$ in terms of other parameters of $G$. 2. Study $f(n,k)$ defined to be the maximum number of edges in a graph on $n$ vertices with $mp(G) < k$. 3. Estimate t…
The eccentricity matrix of a connected graph $G$, denoted by $\mathcal{E}(G)$, is obtained from the distance matrix of $G$ by keeping the largest nonzero entries in each row and each column, and leaving zeros in the remaining ones. The $\mathcal{E}$-eigenvalues of $G$ are the eigenvalues of $\mathcal{E}(G)$, in which the largest one is the $\mathcal{E}$-spectral radius of $G$. The $\mathcal{E}$-energy of $G$ is the sum of the absolute values of all $\mathcal{E}$-eigenvalues of $G$. In this article, we study some of the extremal problems for eccentricity matrices of complements of trees and cha…
A 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 a connected graph $G$ and a given integer $k$ with $2\leq k\leq |V(G)|$, the \emph{$k$-monochromatic index $mx_k(G)$} of $G$ is the maximum number of colors needed such that for each subset $S\subseteq V(G)$ of $k$ vertices, there exists a monochromatic $S$-tree. In this paper, we prove that for any connected graph $G$, $mx_k(G)=|E(G)|-|V(G)|+2$ …
The Omega Institute（欧米伽研究所）；ChronoAI Pte Ltd（ChronoAI私人有限公司）；Department of Fundamental Science and Technology National Higher School of Advanced Technologies（国家高等先进技术学院基础科学与技术系）；National University of Singapore（新加坡国立大学）
