17 research outputs found
Distances and Domination in Graphs
This book presents a compendium of the 10 articles published in the recent Special Issue âDistance and Domination in Graphsâ. The works appearing herein deal with several topics on graph theory that relate to the metric and dominating properties of graphs. The topics of the gathered publications deal with some new open lines of investigations that cover not only graphs, but also digraphs. Different variations in dominating sets or resolving sets are appearing, and a review on some networksâ curvatures is also present
Advances in Discrete Applied Mathematics and Graph Theory
The present reprint contains twelve papers published in the Special Issue âAdvances in Discrete Applied Mathematics and Graph Theory, 2021â of the MDPI Mathematics journal, which cover a wide range of topics connected to the theory and applications of Graph Theory and Discrete Applied Mathematics. The focus of the majority of papers is on recent advances in graph theory and applications in chemical graph theory. In particular, the topics studied include bipartite and multipartite Ramsey numbers, graph coloring and chromatic numbers, several varieties of domination (Double Roman, Quasi-Total Roman, Total 3-Roman) and two graph indices of interest in chemical graph theory (Sombor index, generalized ABC index), as well as hyperspaces of graphs and local inclusive distance vertex irregular graphs
Revolutionaries and spies: Spy-good and spy-bad graphs
We study a game on a graph played by {\it revolutionaries} and
{\it spies}. Initially, revolutionaries and then spies occupy vertices. In each
subsequent round, each revolutionary may move to a neighboring vertex or not
move, and then each spy has the same option. The revolutionaries win if of
them meet at some vertex having no spy (at the end of a round); the spies win
if they can avoid this forever.
Let denote the minimum number of spies needed to win. To
avoid degenerate cases, assume |V(G)|\ge r-m+1\ge\floor{r/m}\ge 1. The easy
bounds are then \floor{r/m}\le \sigma(G,m,r)\le r-m+1. We prove that the
lower bound is sharp when has a rooted spanning tree such that every
edge of not in joins two vertices having the same parent in . As a
consequence, \sigma(G,m,r)\le\gamma(G)\floor{r/m}, where is the
domination number; this bound is nearly sharp when .
For the random graph with constant edge-probability , we obtain constants
and (depending on and ) such that is near the
trivial upper bound when and at most times the trivial lower
bound when . For the hypercube with , we have
when , and for at least spies are
needed.
For complete -partite graphs with partite sets of size at least , the
leading term in is approximately
when . For , we have
\sigma(G,2,r)=\bigl\lceil{\frac{\floor{7r/2}-3}5}\bigr\rceil and
\sigma(G,3,r)=\floor{r/2}, and in general .Comment: 34 pages, 2 figures. The most important changes in this revision are
improvements of the results on hypercubes and random graphs. The proof of the
previous hypercube result has been deleted, but the statement remains because
it is stronger for m<52. In the random graph section we added a spy-strategy
resul
Roman Domination in Complementary Prism Graphs
A Roman domination function on a complementary prism graph GGc is a function f : V [ V c ! {0, 1, 2} such that every vertex with label 0 has a neighbor with label 2. The Roman domination number R(GGc) of a graph G = (V,E) is the minimum of Px2V [V c f(x) over such functions, where the complementary prism GGc of G is graph obtained from disjoint union of G and its complement Gc by adding edges of a perfect matching between corresponding vertices of G and Gc. In this paper, we have investigated few properties of R(GGc) and its relation with other parameters are obtaine
Some Investigations in the Theory of Graphs
Not availabl
International Journal of Mathematical Combinatorics, Vol.6A
The International J.Mathematical Combinatorics (ISSN 1937-1055) is a fully refereed international journal, sponsored by the MADIS of Chinese Academy of Sciences and published in USA quarterly comprising 460 pages approx. per volume, which publishes original research papers and survey articles in all aspects of Smarandache multi-spaces, Smarandache geometries, mathematical combinatorics, non-euclidean geometry and topology and their applications to other sciences
Symmetry in Graph Theory
This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view
Discrete Mathematics and Symmetry
Some of the most beautiful studies in Mathematics are related to Symmetry and Geometry. For this reason, we select here some contributions about such aspects and Discrete Geometry. As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we consider network structures, symmetry means invariance of adjacency of nodes under the permutations of node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group