Relations between Kirchhoff index and Laplacian–energy–like invariant
Relations between the domination parameters and the chromatic index of a graph
In this paper we show upper bounds for the sum and the product of the lower domination parameters and the chromatic index of a graph. We also present some families of graphs for which these upper bounds are achieved. Next, we give a lower bound for the sum of the upper domination parameters and the chromatic index. This lower bound is a function of the number of vertices of a graph and a new graph parameter which is defined here. In this case we also characterize graphs for which a respective equality...
Relations between -regular sets and star complements
Let be a finite graph with an eigenvalue of multiplicity . A set of vertices in is called a star set for in if is not an eigenvalue of the star complement which is the subgraph of induced by vertices not in . A vertex subset of a graph is -regular if it induces a -regular subgraph and every vertex not in the subset has neighbors in it. We investigate the graphs having a -regular set which induces a star complement for some eigenvalue. A survey of known results is provided...
Relations, coverings, hypergraphs and matroids
Relationships between distance domination parameters.
Relative expressiveness of the edge/adjacency language for graph theory
Relaxations of Ore's condition on cycles.
Relaxed graceful labellings of trees.
Reliability of communication networks with delay constraints: computational complexity and complete topologies.
Remark about a construction of some tournaments points of certain projective planes
Remark on a paper of Lovász
Remark on inequalities for the Laplacian spread of graphs
Two inequalities for the Laplacian spread of graphs are proved in this note. These inequalities are reverse to those obtained by Z. You, B. Liu: The Laplacian spread of graphs, Czech. Math. J. 62 (2012), 155–168.
Remark to a problem on 0-1 matrices
Remarks on 15-vertex (3,3)-ramsey graphs not containing K₅
The paper gives an account of previous and recent attempts to determine the order of a smallest graph not containing K₅ and such that every 2-coloring of its edges results in a monochromatic triangle. A new 14-vertex K₄-free graph with the same Ramsey property in the vertex coloring case is found. This yields a new construction of one of the only two known 15-vertex (3,3)-Ramsey graphs not containing K₅.
Remarks on best approximation in R-trees
An R-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. If X is a closed convex subset of an R-tree Y, and if T: X → 2Y is a multivalued mapping, then a point z for which [...] is called a point of best approximation. It is shown here that if T is an ε-semicontinuous mapping whose values are nonempty closed convex subsets of Y, and if T has at least two distinct points of best approximation, then T must have a fixed point. We...
Remarks on -integral complete multipartite graphs
A graph is called distance integral (or -integral) if all eigenvalues of its distance matrix are integers. In their study of -integral complete multipartite graphs, Yang and Wang (2015) posed two questions on the existence of such graphs. We resolve these questions and present some further results on -integral complete multipartite graphs. We give the first known distance integral complete multipartite graphs with , and with , as well as the infinite classes of distance integral complete...
Remarks on diagonalizable embeddings of graphs
Remarks on dimensions of graphs
Remarks on Dynamic Monopolies with Given Average Thresholds
Dynamic monopolies in graphs have been studied as a model for spreading processes within networks. Together with their dual notion, the generalized degenerate sets, they form the immediate generalization of the classical notions of vertex covers and independent sets in a graph. We present results concerning dynamic monopolies in graphs of given average threshold values extending and generalizing previous results of Khoshkhah et al. [On dynamic monopolies of graphs: The average and strict majority...
Remarks on flows in network with short paths