Independence numbers of the 2-token graphs of some join graphs

The independent number and domination number are two essential parameters to assess the resilience of the interconnection network of multiprocessor systems which is usually modeled by a graph. The independent number, denoted by [Formula: see text], of a graph [Formula: see text] is the maximum cardinality of any subset [Formula: see text] such that no two elements in [Formula: see text] are adjacent in [Formula: see text]. The domination number, denoted by [Formula: see text], of a graph [Formula: see text] is the minimum cardinality of any subset [Formula: see text] such that every vertex in [Formula: see text] is either in [Formula: see text] or adjacent to an element of [Formula: see text]. But so far, determining the independent number and domination number of a graph is still an NPC problem. Therefore, it is of utmost importance to determine the number of independent and domination number of some special networks with potential applications in multiprocessor system. In this paper, we firstly resolve the exact values of independent number and upper and lower bound of domination number of the [Formula: see text]-graph, a common generalization of various popular interconnection networks. Besides, as by-products, we derive the independent number and domination number of [Formula: see text]-star graph [Formula: see text], [Formula: see text]-arrangement graph [Formula: see text], as well as three special graphs.

Extending Berge’s and Favaron’s results about well-covered graphs

In this paper, we disprove the claimed characterisation of graphs with equal independence and annihilation number as proposed by Larson and Pepper [Electron. J. Comb. 2011]. The annihilation number of a graph is defined as the largest integer $p$ such that the sum of its smallest $p$ degrees is greater than or equal to its size, i.e., its number of edges. Larson and Pepper claimed that for a given graph $G=(V,E)$, its independence number $\alpha(G)$ equals its annihilation number $a(G)$ if and only if $$\begin{array}{ll}(1)~~ a(G)\geq \frac n2:& \alpha'(G)=a(G)\\[2mm](2)~~ a(G)= \frac{n-1}{2}:& \alpha'(G-v)=a(G) ~\text{ for some } v\in V.\end{array}$$This paper provides series of counterexamples with an arbitrarily large number of vertices, an arbitrarily large number of components, an arbitrarily large independence number, and an arbitrarily large difference between the critical and the regular independence number. Furthermore, we identify the error in the proof of Larson and Pepper's theorem. Yet, we show that the theorem still holds for bipartite graphs and connected claw-free graphs.

Graph theory is a delightful playground for the exploration of proof techniques in discrete mathematics and its results have applications in many areas of the computing, social, and natural sciences. The fastest growing area within graph theory is the study of domination and Independence numbers. Domination number is the cardinality of a minimum dominating set of a graph. Independence number is the maximal cardinality of an independent set of vertices of a graph. The concept of Fibonacci numbers of graphs was first introduced by Prodinger and Tichy in 1982. The Fibonacci numbers of a graph is the number of independent vertex subsets. In this paper, introduce the identities of domination, independence and Fibonacci numbers of graphs containing vertex-disjoint cycles and edge-disjoint cycles.

This paper examines invariants of the replacement product of two graphs in terms of the properties of the component graphs. In particular, we present results on the independence number, the domination number, and the total domination number of these graphs. The replacement product is a noncommutative graph operation that has been widely applied in many areas. One of its advantages over other graph products is its ability to produce sparse graphs. The results in this paper give insight into how to construct large, sparse graphs with optimal independence or domination numbers.

The annihilation number $a$ of a graph is an upper bound of the independence number $\alpha$ of a graph. In this article we characterize graphs with equal independence and annihilation numbers. In particular, we show that $\alpha=a$ if, and only if, either (1) $a\geq \frac{n}{2}$ and $\alpha' =a$, or (2) $a < \frac{n}{2}$ and there is a vertex $v\in V(G)$ such that $\alpha' (G-v)=a(G)$, where $\alpha'$ is the critical independence number of the graph. Furthermore, we show that it can be determined in polynomial time whether $\alpha=a$. Finally we show that a graph where $\alpha=a$ is either König-Egerváry or almost König-Egerváry.

The independent domination number i(G) (independent number β(G)) is the minimum (maximum) cardinality among all maximal independent sets of G. Haviland (1995) conjectured that any connected regular graph G of order n and degree δ ≤ 1/2n satisfies i(G) ≤ ⌈2n/3δ⌉ 1/2δ. For 1 ≤ k ≤ l ≤ m, the subset graph S m (k, l) is the bipartite graph whose vertices are the k- and l-subsets of an m element ground set where two vertices are adjacent if and only if one subset is contained in the other. In this paper, we give a sharp upper bound for i(S m (k, l)) and prove that if k + l = m then Haviland’s conjecture holds for the subset graph S m (k, l). Furthermore, we give the exact value of β(S m (k, l)).

Let [Formula: see text] be a graph of order [Formula: see text]. Let [Formula: see text] and [Formula: see text] denote respectively the independent domination number, independence number and chromatic number of [Formula: see text] These parameters satisfy the well known inequality [Formula: see text]. In this paper we determine conditions under which the above inequalities become equalities and use these conditions to determine extremal graphs in some specific graph classes.

For a graph G, let n(G), \(\alpha (G)\) and \(\beta (G)\) be its order, independence number and matching number, respectively. We showed that \(\frac{\Delta (G)+k}{4}\alpha (G) + \beta (G) \ge n(G)\) for some \(K_k\)-free graph G with \(\Delta (G)\ge k-1\ge 2\).

In this chapter, we study Vizing’s Conjecture from 1968 which asserts that the domination number of the Cartesian product of two graphs is at least as large as the product of their domination numbers. The conjecture was first posed by Vizing as a question in 1963. Vizing’s Conjecture is considered by many to be the main open problem in the area of domination in graphs. We also present Vizing-like conjectures for the total domination number, the independent domination number, the independence number, the upper domination number, and the upper total domination number in Cartesian products of graphs.

Index of parameters of iterated line graphs

In the present paper, we obtain bounds for Harary index $H(G)$ of a connected (molecular) graph in terms of vertex connectivity, independent number, independent domination number and characterize graphs extremal with respect to them.

Let $G$ be a simple graph of order $n$ and $\mathscr{L}(G) \equiv \mathscr{L}^{1}(G)$ its line graph. Then, the iterated line graph of $G$ is defined recursively as $\mathscr{L}^{2}(G) \equiv \mathscr{L}(\mathscr{L}(G)), \mathscr{L}^{3}(G)\equiv \mathscr{L}(\mathscr{L}^{2}(G)), \ldots, \mathscr{L}^{k}(G)\equiv\mathscr{L}\left(\mathscr{L}^{k-1}(G)\right).$ The energy $\mathcal{E}(G)$ is the sum of absolute values of the eigenvalues of $G$. In this paper, it is derived a sharp upper bound for the energy of the line graph of a connected graph $G$ of order $n$ and independence number not less than $\alpha$ where $1\leq\alpha\leq n-2$. This bound is attained, if and only if, $G$ is isomorphic to the complete split graphs $SK_{n,\alpha}$. It is also determined a lower bound for the energy of the line graph of a graph $G$ of order $n$ and independence number $\alpha$. For $1\leq\alpha\leq n-1$ and $\mathcal{H}=\left(n-\alpha\left\lfloor\dfrac{n}{\alpha}\right\rfloor\right)K_{\lfloor\frac{n}{\alpha}\rfloor+1}\bigcup \left(\alpha+\alpha\left\lfloor\dfrac{n}{\alpha}\right\rfloor-n\right)K_{\lfloor\frac{n}{\alpha}\rfloor}$, the equality holds, if and only if $G \cong \mathcal{H}.$ As a consequence, families of hyperenergetic graphs are determined. Also, a lower bound for the energy of the iterated line of a graph $G$ of order $n$ and independence number $\alpha$ is given and, for $1\leq\alpha\leq n-1$, the equality holds, if and only if, $G\cong \alpha K_{\left\lfloor\frac{n}{\alpha}\right\rfloor}$. Additionally, an upper bound for the incidence energy of connected graphs $G$ of order $n$ and independence number not less than $\alpha$ is presented. Moreover, an upper bound on the Laplacian energy-like of the complement $\overline{G}$ of $G$ is presented. For $1\leq\alpha\leq n-1$, the bound is attained, if and only if, $G\cong \mathcal{H}.$ Finally, a Nordhaus-Gaddum type relation is given.

formula omitted]-perfect graphs

Independence Saturation and Extended Domination Chain in Graphs