Nordhaus Gaddum-type Inequality Research Articles

Laplacian coefficients, Kirchhoff index and the number of spanning trees of graphs

Let [Formula: see text], [Formula: see text], be a simple connected graph of order [Formula: see text] and size [Formula: see text]. Denote with [Formula: see text] eigenvalues of the Laplacian matrix [Formula: see text] of [Formula: see text]. The Kirchhoff index and the number of spanning trees of [Formula: see text] expressed in terms of Laplacian eigenvalues are given by [Formula: see text] and [Formula: see text], respectively. The characteristic polynomial of [Formula: see text] is given by [Formula: see text]. The first five Laplacian coefficients have been computed in the literature. In this study, we compute the sixth Laplacian coefficient of [Formula: see text]. Then, we use it to improve the previously obtained results on [Formula: see text] and [Formula: see text]. In addition, we present new Nordhaus–Gaddum-type inequalities for [Formula: see text] and [Formula: see text].

Equitable and Paired Equitable Domination in Inflated Graphs and Their Complements

Domination plays an indispensable role in graph theory. Various types of domination explore various types of applications. Equal-status people work together and interlace with each other easily. In this paper, the paired equitable domination of a graph, its inflated graph, and its complement of an inflated graph were studied. The relationship between the domination number of the graph, the equitable domination number, and the paired equitable domination number of complements of the inflated graph were established. Furthermore, we proved the Nordhaus–Gaddum-type inequality, that is, γpre(H)+γpre(H)≤6 if H is a graph with m nodes where m≡0, 2(mod 8) and d(ai) = (m/2) for all ai. The challenges and limitations of this parameter of paired equitable and equitable domination depends on the degree of the vertex of the graph. Practical applications are discussed in various fields and illustrated using the studied parameter.

New Improved Bounds for Signless Laplacian Spectral Radius and Nordhaus-Gaddum Type Inequalities for Agave Class of Graphs

Core-satellite graphs Θ(c, s, η) ∼= Kc ▽ (ηKs) are graphs consisting of a central clique Kc (the core) and η copies of Ks (the satellites) meeting in a common clique. They belong to the class of graphs of diameter two. Agave graphs Θ(2, 1, η) ∼= K2 ▽ (ηK1) belong to the general class of complete split graphs, where the graphs consist of a central clique K2 and η copies of K1 which are connected to all the nodes of the clique. They are the subclass of Core-satellite graphs. Let μ(G) be the spectral radius of the signless Laplacian matrix Q(G). In this paper, we have obtained the greatest lower bound and the least upper bound of signless Laplacian spectral radius of Agave graphs. These bounds have been expressed in terms of graph invariants like m the number of edges, n the number of vertices, δ the minimum degree, ∆ the maximum degree, and η copies of the satellite. We have made use of the approximation technique to derive these bounds. This unique approach can be utilized to determine the bounds for the signless Laplacian spectral radius of any general family of graphs. We have also obtained Nordhaus-Gaddum type inequality using the derived bounds.

Nordhaus–Gaddum type inequality for the integer k-matching number of a graph

An integer k-matching of a graph G is a function h:E(G)→{0,1,…,k} such that ∑e∈Γ(v)h(e)≤k for any v∈V(G), where Γ(v) is the set of edges incident to v. The integer k-matching number of G, denoted by mk(G), is the maximum number of ∑e∈E(G)h(e) over all integer k-matching h of G. In this paper, we establish the following lower bounds on the sum of the integer k-matching number of a graph G and its complement by using Gallai–Edmonds Structure Theorem: (1) mk(G)+mk(G¯)≥⌊nk2⌋ for n≥2;(2) if G and G¯ are non-empty, then for n≥25, mk(G)+mk(G¯)≥⌊nk+k2⌋;(3) if G and G¯ have no isolated vertices, then for n≥25, mk(G)+mk(G¯)≥⌊nk2⌋+2k . Furthermore, all extremal graphs attaining the lower bounds are also characterized.

New bounds on the outer-independent total double Roman domination number

A double Roman dominating function (DRDF) on a graph [Formula: see text] is a function [Formula: see text] satisfying (i) if [Formula: see text] then there must be at least two neighbors assigned two under [Formula: see text] or one neighbor [Formula: see text] with [Formula: see text]; and (ii) if [Formula: see text] then [Formula: see text] must be adjacent to a vertex [Formula: see text] such that [Formula: see text]. A DRDF is an outer-independent total double Roman dominating function (OITDRDF) on [Formula: see text] if the set of vertices labeled [Formula: see text] induces an edgeless subgraph and the subgraph induced by the vertices with a non-zero label has no isolated vertices. The weight of an OITDRDF is the sum of its function values over all vertices, and the outer-independent total Roman domination number [Formula: see text] is the minimum weight of an OITDRDF on [Formula: see text]. In this paper, we establish various bounds on [Formula: see text]. In particular, we present Nordhaus–Gaddum-type inequalities for this parameter. Some of our results improve the previous results.

Nordhaus–Gaddum type inequalities on the total Italian domination number in graphs

Let G be a graph with vertex set V(G). A total Italian dominating function (TIDF) on a graph G is a function f : V(G) → {0, 1, 2} such that (i) every vertex v with f(v) = 0 is adjacent to a vertex u with f(u) = 2 or to two vertices w and z with f(w) = f(z) = 1, and (ii) every vertex v with f(v) ≥ 1 is adjacent to a vertex u with f(u) ≥ 1. The total Italian domination number γtI(G) on a graph G is the minimum weight of a total Italian dominating function. In this paper, we present Nordhaus–Gaddum type inequalities for the total Italian domination number.

On spectral radius and Nordhaus-Gaddum type inequalities of the generalized distance matrix of graphs

If $Tr(G)$ and $D(G)$ are respectively the diagonal matrix of vertex transmission degrees and distance matrix of a connected graph $G$, the generalized distance matrix $D_{\alpha}(G)$ is defined as $D_{\alpha}(G)=\alpha ~Tr(G)+(1-\alpha)~D(G)$, where $0\leq \alpha \leq 1$. If $\rho_1 \geq \rho_2 \geq \dots \geq \rho_n$ are the eigenvalues of $D_{\alpha}(G)$, the largest eigenvalue $\rho_1$ (or $\rho_{\alpha}(G)$) is called the spectral radius of the generalized distance matrix $D_{\alpha}(G)$. The generalized distance energy is defined as $E^{D_{\alpha}}(G)=\sum_{i=1}^{n}\left|\rho_i -\frac{2\alpha W(G)}{n}\right|$, where $W(G)$ is the Wiener index of $G$. In this paper, we obtain the bounds for the spectral radius $\rho_{\alpha}(G)$ and the generalized distance energy of $G$ involving Wiener index. We derive the Nordhaus-Gaddum type inequalities for the spectral radius and the generalized distance energy of $G$.

Nordhaus-gaddum type inequalities for tree covering numbers on unitary cayley graphs of finite rings

The unitary Cayley graph $Gamma_n$ of a finite ring $mathbb{Z}_n$ is the graph with vertex set $mathbb{Z}_n$ and two vertices $x$ and $y$ are adjacent if and only if $x-y$ is a unit in $mathbb{Z}_n$‎. ‎A family $mathcal{F}$ of mutually edge disjoint trees in $Gamma_n$ is called a tree cover of $Gamma_n$ if for each edge $ein E(Gamma_n)$‎, ‎there exists a tree $Tinmathcal{F}$ in which $ein E(T)$‎. ‎The minimum cardinality among tree covers of $Gamma_n$ is called a tree covering number and denoted by $tau(Gamma_n)$‎. ‎In this paper‎, ‎we prove that‎, ‎for a positive integer $ ngeq 3 $‎, ‎the tree covering number of $ Gamma_n $ is $ displaystylefrac{varphi(n)}{2}+1 $ and the tree covering number of $ overline{Gamma}_n $ is at most $ n-p $ where $ p $ is the least prime divisor of $n$‎. ‎Furthermore‎, ‎we introduce the Nordhaus-Gaddum type inequalities for tree covering numbers on unitary Cayley graphs of rings $mathbb{Z}_n$‎.

Nordhaus–Gaddum type inequality for the fractional matching number of a graph

The fractional matching number of a graph G, written as α′(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) α′(G)+α′(G¯)≥n2 for n≥2.(2) If G and G¯ are non-empty, then for n≥30, α′(G)+α′(G¯)≥n+12.(3) If G and G¯ have no isolated vertices, then for n≥30, α′(G)+α′(G¯)≥n+42.

Regular graphs with large Italian domatic number

For a graph $G$, an Italian dominating function is a function $f: V(G) rightarrow {0,1,2}$ such that for each vertex $v in V(G)$ either $f(v) neq 0$, or $sum_{u in N(v)} f(u) geq 2$.If a family $mathcal{F} = {f_1, f_2, dots, f_t}$ of distinct Italian dominating functions satisfy $sum^t_{i = 1} f_i(v) leq 2$ for each vertex $v$, then this is called an Italian dominating family.In [L. Volkmann, The {R}oman {${2}$}-domatic number of graphs, Discrete Appl. Math. {bf 258} (2019), 235--241], Volkmann defined the textit{Italian domatic number} of $G$, $d_{I}(G)$, as the maximum cardinality of any Italian dominating family. In this same paper, questions were raised about the Italian domatic number of regular graphs. In this paper, we show that two of the conjectures are false, and examine some exceptions to a Nordhaus-Gaddum type inequality.

On three outer-independent domination related parameters in graphs

Let G be a graph and let S⊆V(G). The set S is a double outer-independent dominating set of G if |N[v]∩S|≥2 for all v∈V(G), and V(G)∖S is independent. Similarly, S is a 2-outer-independent dominating set, if every vertex from V(G)∖S has at least two neighbors in S and V(G)∖S is independent. Finally, S is a total outer-independent dominating set if every vertex from V(G) has a neighbor in S and the complement of S is an independent set. The double, total or 2-outer-independent domination number of G is the smallest possible cardinality of any double, total or 2-outer-independent dominating set of G, respectively. In this paper, the 2-outer-independent, the total outer-independent and the double outer-independent domination numbers of graphs are investigated. We prove some Nordhaus–Gaddum type inequalities, derive their computational complexities and present several bounds for them.

The Connectivity of a Bipartite Graph and Its Bipartite Complementary Graph

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 [Formula: see text] and the same invariant in the complement [Formula: see text] of [Formula: see text] 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 [Formula: see text] with bipartition ([Formula: see text]), its bipartite complementary graph [Formula: see text] is a bipartite graph with [Formula: see text] and [Formula: see text] and [Formula: see text]. In this paper, we obtain the Nordhaus-Gaddum type inequalities for connectivity of bipartite graphs and its bipartite complementary graphs. Furthermore, we prove that these inequalities are best possible.

Nordhaus–Gaddum type inequalities for some distance-based indices of bipartite molecular graphs

Nordhaus–Gaddum type inequalities for some distance-based indices of bipartite molecular graphs

Nordhaus-Gaddum type inequalities of the second Aα-eigenvalue of a graph

Let G be a graph with adjacency matrix A(G) and the degree diagonal matrix D(G). For any real α∈[0,1], the matrix Aα(G) of a graph is defined as Aα(G)=αD(G)+(1−α)A(G). Denote its eigenvalues by λ1(Aα(G))≥λ2(Aα(G))≥⋯≥λn(Aα(G)). In this paper, for 1/2<α<1, the Nordhaus-Gaddum type bound for the second largest Aα-eigenvalue of a graph is considered. We show that λ2(Aα(G))+λ2(Aα(G‾))≥nα−1, and the extremal graphs for which the equality holds are Kn and Sn. If G∉{Kn,Sn,Kn−e}, then λ2(Aα(G))+λ2(Aα(G‾))≥(n−1)α. Moreover, we give two upper Nordhaus-Gaddum type bounds for the second largest Aα-eigenvalue of a graph.

Extremal graphs for Estrada indices

Let G be a simple undirected connected graph. The signless Laplacian Estrada, Laplacian Estrada and Estrada indices of a graph G is the sum of the exponentials of the signless Laplacian eigenvalues, Laplacian eigenvalues and eigenvalues of G, respectively. The present work derives an upper bound for the Estrada index of a graph as a function of its chromatic number, in the family of graphs whose color classes have order not less than a fixed positive integer. The graphs for which the upper bound is tight is obtained. Additionally, an upper bound for the Estrada Index of the complement of a graph in the previous family of graphs with two color classes is given. A Nordhaus-Gaddum type inequality for the Laplacian Estrada index when G is a bipartite graph with color classes of order not less than 2, is presented. Moreover, a sharp upper bound for the Estrada index of the line graph and for the signless Laplacian index of a graph in terms of connectivity is obtained.

Nordhaus–Gaddum and other bounds for the chromatic edge-stability number

Let G be a graph. The chromatic edge-stability number esχ(G) of a graph G is the minimum number of edges of G whose removal results in a graph H with χ(H)=χ(G)−1. A Nordhaus–Gaddum type inequality for the chromatic edge-stability number is proved. Sharp upper bounds on esχ are given for general graphs in terms of size and of maximum degree, respectively. All bounds are demonstrated to be sharp. Graphs with esχ=1 are considered and in particular characterized among k-regular graphs for k≤5. Several open problems are also stated.

The Nordhaus–Gaddum type inequalities of Aα-matrix

For a real number α ∈ [0, 1], the Aα-matrix of a graph G is defined as Aα(G)=αD(G)+(1−α)A(G), where A(G) and D(G) are the adjacency matrix and diagonal degree matrix of G, respectively. The Aα-spectral radius of G, denoted by ρα(G), is the largest eigenvalue of Aα(G). In this paper, the Nordhaus–Gaddum type bounds for the Aα-spectral radius are considered.

Packing parameters in graphs: new bounds and a solution to an open problem

In this paper, we investigate the packing parameters in graphs. By applying the Mantel's theorem, We give upper bounds on packing and open packing numbers of triangle-free graphs along with characterizing the graphs for which the equalities hold and exhibit sharp Nordhaus-Gaddum type inequalities for packing numbers. We also solve the open problem of characterizing all connected graphs with $\rho_{o}(G)=n-\omega(G)$ posed in [S. Hamid and S. Saravanakumar, {\em Packing parameters in graphs}, Discuss Math. Graph Theory, 35 (2015), 5--16].

Nordhaus–Gaddum type inequalities for the two largest Laplacian eigenvalues

For a simple graph G with n vertices, let μ1(G)≥μ2(G)≥⋯≥μn−1(G)≥μn(G)=0 be the Laplacian eigenvalues of G. We study Nordhaus–Gaddum type inequalities for μ1(G) and μ2(G). We improve some existing results from the literature for μ1(G) and obtain new results for μ2(G). We also propose a conjecture that μ2(G)+μ2(G¯)≤2n−2holds for any graph G and prove it is true when G or G¯ is disconnected; when G is bipartite; for regular graphs and when G and G¯ have diameter not equal to 2.

On Nordhaus–Gaddum type inequalities for the game chromatic and game coloring numbers

A seminal result by Nordhaus and Gaddum states that 2n≤χ(G)+χ(G¯)≤n+1 for every graph G of order n, where G¯ is the complement of G and χ is the chromatic number. We study similar inequalities for χg(G) and colg(G), which denote, respectively, the game chromatic number and the game coloring number of G. Those graph invariants give the score for, respectively, the coloring and marking games on G when both players use their best strategies.