The Impact of Edge Correlations in Random Networks

Random graphs are frequently used models of real-life random networks. The classical Erdös–Rényi random graph model is very well explored and has numerous nontrivial properties. In particular, a good number of important graph parameters that are hard to compute in the deterministic case often become much easier in random graphs. However, a fundamental restriction in the Erdös–Rényi random graph is that the edges are required to be probabilistically independent. This is a severe restriction, which does not hold in most real-life networks. We consider more general random graphs in which the edges may be dependent. Specifically, two models are analyzed. The first one is called a p-robust random graph. It is defined by the requirement that each edge exist with probability at least p, no matter how we condition on the presence/absence of other edges. It is significantly more general than assuming independent edges existing with probability p, as exemplified via several special cases. The second model considers the case when the edges are positively correlated, which means that the edge probability is at least p for each edge, no matter how we condition on the presence of other edges (but absence is not considered). We prove some interesting, nontrivial properties about both models.

On Independent Secondary Dominating Sets in Generalized Graph Products

In 2008, Hedetniemi et al. introduced (1,k)-domination in graphs. The research on this concept was extended to the problem of existence of independent (1,k)-dominating sets, which is an NP-complete problem. In this paper, we consider independent (1,1)- and (1,2)-dominating sets, which we name as (1,1)-kernels and (1,2)-kernels, respectively. We obtain a complete characterization of generalized corona of graphs and G-join of graphs, which have such kernels. Moreover, we determine some graph parameters related to these sets, such as the number and the cardinality. In general, graph products considered in this paper have an asymmetric structure, contrary to other many well-known graph products (Cartesian, tensor, strong).

Energy of extended bipartite double graphs

The energy of a graph is the sum of the absolute values of its eigenvalues. In this article, an exact relation between the energy of extended bipartite double graph and the energy of a graph together with some other graph parameters is given. As a consequence, equienergetic, borderenergetic, orderenergetic and non-hyperenergetic extended bipartite double graphs are presented. The obtained results generalize the existing results on equienergetic bipartite graphs.

Relations between the energy of graphs and other graph parameters

In this paper we give various relations between the energy of graphs and other graph parameters as Randić index, clique number, number of vertices and edges, maximum and minimum degree etc. Moreover, new bounds for the energy of complementary graphs are derived. Our results are based on the concept of vertex energy developed by G. Arizmendi and O. Arizmendi in [Lin. Algebra Appl. doi:10.1016/j.laa.2020.09.025].

Low-congestion shortcut and graph parameters

Distributed graph algorithms in the standard CONGEST model often exhibit the time-complexity lower bound of $${\tilde{\Omega }}(\sqrt{n} + D)$$ Ω ~ ( n + D ) rounds for several global problems, where n denotes the number of nodes and D the diameter of the input graph. Because such a lower bound is derived from special “hard-core” instances, it does not necessarily apply to specific popular graph classes such as planar graphs. The concept of low-congestion shortcuts was initiated by Ghaffari and Haeupler [SODA2016] for addressing the design of CONGEST algorithms running fast in restricted network topologies. In particular, given a graph class $${\mathcal {C}}$$ C , an f-round algorithm for constructing shortcuts of quality q for any instance in $${\mathcal {C}}$$ C results in $${\tilde{O}}(q + f)$$ O ~ ( q + f ) -round algorithms for solving several fundamental graph problems such as minimum spanning tree and minimum cut, for $${\mathcal {C}}$$ C . The main interest on this line is to identify the graph classes allowing the shortcuts that are efficient in the sense of breaking $${\tilde{O}}(\sqrt{n}+D)$$ O ~ ( n + D ) -round general lower bounds. In this study, we consider the relationship between the quality of low-congestion shortcuts and the following four major graph parameters: doubling dimension, chordality, diameter, and clique-width. The key ingredient of the upper-bound side is a novel shortcut construction technique known as short-hop extension, which might be of independent interest.

On the 2-Packing Differential of a Graph

Let G be a graph of order $${\text {n}}(G)$$ n ( G ) and vertex set V(G). Given a set $$S\subseteq V(G)$$ S ⊆ V ( G ) , we define the external neighbourhood of S as the set $$N_e(S)$$ N e ( S ) of all vertices in $$V(G){\setminus } S$$ V ( G ) \ S having at least one neighbour in S. The differential of S is defined to be $$\partial (S)=|N_e(S)|-|S|$$ ∂ ( S ) = | N e ( S ) | - | S | . In this paper, we introduce the study of the 2-packing differential of a graph, which we define as $$\partial _{2p}(G)=\max \{\partial (S):\, S\subseteq V(G) \text { is a }2\text {-packing}\}.$$ ∂ 2 p ( G ) = max { ∂ ( S ) : S ⊆ V ( G ) is a 2 -packing } . We show that the 2-packing differential is closely related to several graph parameters, including the packing number, the independent domination number, the total domination number, the perfect differential, and the unique response Roman domination number. In particular, we show that the theory of 2-packing differentials is an appropriate framework to investigate the unique response Roman domination number of a graph without the use of functions. Among other results, we obtain a Gallai-type theorem, which states that $$\partial _{2p}(G)+\mu _{_R}(G)={\text {n}}(G)$$ ∂ 2 p ( G ) + μ R ( G ) = n ( G ) , where $$\mu _{_R}(G)$$ μ R ( G ) denotes the unique response Roman domination number of G. As a consequence of the study, we derive several combinatorial results on $$\mu _{_R}(G)$$ μ R ( G ) , and we show that the problem of finding this parameter is NP-hard. In addition, the particular case of lexicographic product graphs is discussed.

On the Maximal Shortest Paths Cover Number

A shortest path P of a graph G is maximal if P is not contained as a subpath in any other shortest path. A set S⊆V(G) is a maximal shortest paths cover if every maximal shortest path of G contains a vertex of S. The minimum cardinality of a maximal shortest paths cover is called the maximal shortest paths cover number and is denoted by ξ(G). We show that it is NP-hard to determine ξ(G). We establish a connection between ξ(G) and several other graph parameters. We present a linear time algorithm that computes exact value for ξ(T) of a tree T.

Bounds and extremal graphs for Harary energy

Let [Formula: see text] be a connected graph of order [Formula: see text] and let [Formula: see text] be the reciprocal distance matrix (also called Harary matrix) of the graph [Formula: see text]. Let [Formula: see text] be the eigenvalues of the reciprocal distance matrix [Formula: see text] of the connected graph [Formula: see text] called the reciprocal distance eigenvalues of [Formula: see text]. The Harary energy [Formula: see text] of a connected graph [Formula: see text] is defined as sum of the absolute values of the reciprocal distance eigenvalues of [Formula: see text], that is, [Formula: see text] In this paper, we establish some new lower and upper bounds for [Formula: see text] in terms of different graph parameters associated with the structure of the graph [Formula: see text]. We characterize the extremal graphs attaining these bounds. We also obtain a relation between the Harary energy and the sum of [Formula: see text] largest adjacency eigenvalues of a connected graph.