On the restrained domination stability in graphs

The toughness of Kneser graphs

In 2012, Ponraj et al. defined k-product cordial labeling as follows: Let [Formula: see text] be a map from [Formula: see text] to [Formula: see text] where [Formula: see text] is an integer, [Formula: see text]. For each edge [Formula: see text] assign the label [Formula: see text] [Formula: see text]. [Formula: see text] is called a k-product cordial labeling if [Formula: see text], and [Formula: see text], [Formula: see text], where [Formula: see text] and [Formula: see text] denote the number of vertices and edges, respectively, labeled with [Formula: see text] [Formula: see text]. A graph that admits k-product cordial labeling is called k-product cordial graph. Later, we proved that several families of graphs are k-product cordial graphs. In this paper, we show that the product of graphs admit k-product cordial labeling.

On the Metric Dimension of Infinite Graphs

The linear relation between Kemeny’s constant, a graph metric directly linked with random walks, and the effective graph resistance in a regular graph has been an incentive to calculate Kemeny’s constant for various networks. In this paper we consider complete bipartite graphs, (generalized) windmill graphs and tree networks with large diameter and give exact expressions of Kemeny’s constant. For non-regular graphs we propose two approximations for Kemeny’s constant by adding to the effective graph resistance term a linear term related to the degree heterogeneity in the graph. These approximations are exact for complete bipartite graphs, but show some discrepancies for generalized windmill and tree graphs. However, we show that a recently obtained upper-bound for Kemeny’s constant in Wang et al. (2017) based on the pseudo inverse Laplacian gives the exact value of Kemeny’s constant for generalized windmill graphs. Finally, we have evaluated Kemeny’s constant, its two approximations and its upper bound, for 243 real-world networks. This evaluation reveals that the upper bound is tight, with average relative error of only 0.73%. In most cases the upper bound clearly outperforms the other two approximations.

Let S be a subset of the vertex set V(G) of a nontrivial connected graph G. The geodetic closure (S) of S is the set of all vertices on geodesics between two vertices in S. The first player A chooses a vertex v1 of G. The second player B then picks v2 ≠ v1 and forms the geodetic closure (S2) = ({v1, v2}). Now A selects v3 e V—S2 and forms (S3) = ({v1, v2, v3}), etc. The player who first selects a vertex vn such that (Sn) = V wins the achievement game, but loses the avoidance game. These geodetic achievement and avoidance games are solved for several families of graphs by determining which player is the winner.

Motivated by the question of how macromolecules assemble,the notion of an assembly tree of a graph is introduced. Given a graph $G$, the paper is concerned with enumerating the number of assembly trees of $G$, a problem that applies to the macromolecular assembly problem. Explicit formulas or generating functions are provided for the number of assembly trees of several families of graphs, in particular for what we call $(H,\phi)$-graphs. In some natural special cases, we use a powerful recent result of Zeilberger and Apagodu to provide recurrence relations for the diagonal of the relevant multivariate generating functions, and we use a result of Wimp and Zeilberger to find very precise asymptotic formulae for the coefficients of these diagonals. Future directions for reseach, as well as open questions, are suggested.

For a graph –(I−Δ)α(12<α≤1), a bijection f from V(G)∪​E(G) to { 1,2,⋯,| V(G) |+| E(G) | } is called an edge-magic labeling of Θ(x) if f(u)+f(v)+f(uv) is independent on the choice of the edge uv. An edge-magic labeling is called super edge-magic if f(V(G))={ 1,2,⋯,| V(G) | }. A graph G is called edge-magic (resp. super edge-magic) if there exists an edge-magic (resp. super edge-magic) labeling of G. In this paper, we investigate whether several families of graphs are (super) edge-magic or not. We also give several conjectures.

We construct and analyze an explicit basis for the homology of the boolean complex of a finite simple graph. This provides a combinatorial description of the spheres in the wedge sum representing the homotopy type of the complex. We assign a set of derangements to any finite simple graph. For each derangement, we construct a corresponding element in the homology of the complex, and the collection of these elements forms a basis for the homology of the boolean complex. In this manner, the spheres in the wedge sum describing the homotopy type of the complex can be represented by a set of derangements. We give an explicit, closed-form description of the derangements that can be obtained from any finite simple graph, and compute this set for several families of graphs. In the cases of complete graphs and Ferrers graphs, these calculations give bijective proofs of previously obtained enumerative results.

This paper is devoted to study the combinatorial properties of Local Minimum Spanning Trees (LMSTs), a geometric structure that is attracting increasing research interest in the wireless sensor networks community. Namely, we study which topologies are allowed for a sensor network that uses, for supporting connectivity, a local minimum spanning tree approach. First, we refine the current definition of LMST realizability, focusing on the role of the power of transmission (i.e., of the radius of the covered area). Second, we show simple planar, connected, and triangle-free graphs with maximum degree 3 that cannot be represented as an LMST. Third, we present several families of graphs that can be represented as LMSTs. Then, we show a relationship between planar graphs and their representability as LMSTs based on homeomorphism. Finally, we show that the general problem of determining whether a graph is LMST representable is NP-hard.

We introduce a notion of curvature on finite, combinatorial graphs. It can be easily computed by solving a linear system of equations. We show that graphs with curvature bounded below by have diameter bounded by (a Bonnet–Myers theorem), that implies that has constant curvature (a Cheng theorem) and that there is a spectral gap (a Lichnerowicz theorem). It is computed for several families of graphs and often coincides with Ollivier curvature or Lin–Lu–Yau curvature. The von Neumann Minimax theorem features prominently in the proofs.

Graphs that allow all the eigenvalue multiplicities to be even

The Sombor index ( SO {\mathrm{SO}} ) is a recently introduced degree-based graph invariant, defined as the sum over all pairs of adjacent vertices u , v {u,v} of the term d u 2 + d v 2 {\sqrt{{d_{u}^{2}+d_{v}^{2}}}} , where d u {d_{u}} and d v {d_{v}} denote the degrees of vertices u and v, respectively. The matrix associated with SO {\mathrm{SO}} is the Sombor matrix, and its spectrum is the Sombor spectrum. In this paper, the connected graphs having exactly two and exactly three Sombor eigenvalues are characterized. Bounds are obtained for the spectral radius and energy of the Sombor matrix, and the corresponding extremal graphs are determined. In addition, the Sombor spectra of several families of graphs are calculated.

There are many practical applications that require simplification of polylines. Some of the goals are to reduce the amount of information necessary to store, improve processing time, or simplify editing. The simplification is usually done by removing some of the vertices, making the resultant polyline go through a subset of the source polyline vertices. However, such approaches do not necessarily produce a new polyline with the minimum number of vertices. The approximate solution to find a polyline, within a specified tolerance, with the minimum number of vertices is described in this paper.

Constructions of bi-regular cages

On the order of [formula omitted]-cages of even girth