Cartesian Product Of Complete Graphs Research Articles

Transitive path decompositions of Cartesian products of complete graphs

Transitive path decompositions of Cartesian products of complete graphs

Lower (total) mutual-visibility number in graphs

Given a graph G, a set X of vertices in G satisfying that between every two vertices in X (respectively, in G) there is a shortest path whose internal vertices are not in X is a mutual-visibility (respectively, total mutual-visibility) set in G. The cardinality of a largest (total) mutual-visibility set in G is known under the name (total) mutual-visibility number, and has been studied in several recent works.In this paper, we propose two lower variants of these concepts, defined as the smallest possible cardinality among all maximal (total) mutual-visibility sets in G, and denote them by μ−(G) and μt−(G), respectively. While the total mutual-visibility number is never larger than the mutual-visibility number in a graph G, we prove that both differences μ−(G)−μt−(G) and μt−(G)−μ−(G) can be arbitrarily large. We characterize graphs G with some small values of μ−(G) and μt−(G), and prove a useful tool called the Neighborhood Lemma, which enables us to find upper bounds on the lower mutual-visibility number in several classes of graphs. We compare the lower mutual-visibility number with the lower general position number, and find a close relationship with the Bollobás-Wessel theorem when this number is considered in Cartesian products of complete graphs. Finally, we also prove the NP-completeness of the decision problem related to μt−(G).

Concentration phenomenon about h-extra edge-connectivity of the n-th cartesian product of complete graph $$K_4$$ with large-scale faulty links

Concentration phenomenon about h-extra edge-connectivity of the n-th cartesian product of complete graph $$K_4$$ with large-scale faulty links

Bakry–Émery Curvature Sharpness and Curvature Flow in Finite Weighted Graphs. Implementation

In this paper, we discuss the implementation of a curvature flow on weighted graphs based on the Bakry–Émery calculus. This flow can be adapted to preserve the Markovian property and its limits as time goes to infinity turn out to be curvature sharp weighted graphs. After reviewing some of the main results of the corresponding paper concerned with the theoretical aspects, we present various examples (random graphs, paths, cycles, complete graphs, wedge sums and Cartesian products of complete graphs, and hypercubes) and exhibit various properties of this flow. One particular aspect of our investigations is asymptotic stability and instability of curvature flow equilibria. The paper ends with a description of the Python functions and routines freely available in an ancillary file on arXiv or via github. We hope that the explanations of the Python implementation via examples will help users to carry out their own curvature flow experiments.

Connection Number- Based Topological Indices of Cartesian Product of Graphs

The area of graph theory (GT) is rapidly expanding and playing a significant role in cheminformatics, mostly in mathematics and chemistry to develop different physicochemical, chemical structure, and their properties. The manipulation and study of chemical graphical details are made feasible by using numerical structure invariant. Investigating these chemical characteristics of topological indices (TIs) is made possible by the discipline of mathematical chemistry. In this article, we study with the Cartesian product of complete graphs, with path graphs, and find their general result of connection number (CN)-based TIs, namely, first connection- based Zagreb index (1st CBZI), second connection- based Zagreb index (2nd CBZI), and third CBZI (3rd CBZI) and then modified first connection- based Zagreb index (CBZI) and second and third modified CBZIs. We also express the general results of first multiplicative CBZI, second multiplicative CBZI, and third and fourth multiplicative CBZI, of two special types of graphs, namely, complete graphs and path graphs. More precisely, we arrange the graphical and numerical analysis of our calculated expressions for both of Cartesian product with each other.

On r-hued coloring of product graphs

A (k,r)-coloring of a graph G is a proper coloring with k colors such that for every vertex v with degree d(v) in G, the color number of the neighbors of v is at least min{d(v),r}. The smallest integer k such that G has a (k,r)-coloring is called the r-hued chromatic number and denoted by χr(G). In Kaliraj et al. [Taibah Univ. Sci. 14 (2020) 168–171], it is determined the 2-hued chromatic numbers of Cartesian product of complete graph and star graph. In this paper, we extend its result and determine the r-hued chromatic number of Cartesian product of complete graph and star graph.

Linkedness of Cartesian products of complete graphs

This paper is concerned with the linkedness of Cartesian products of complete graphs. A graph with at least 2 k vertices is k -linked if, for every set of 2 k distinct vertices organised in arbitrary k pairs of vertices, there are k vertex-disjoint paths joining the vertices in the pairs. We show that the Cartesian product K d 1 +1 × K d 2 +1 of complete graphs K d 1 +1 and K d 2 +1 is ⌊( d 1 + d 2 )/2⌋ -linked for d 1 , d 2 ≥ 2 , and this is best possible. This result is connected to graphs of simple polytopes. The Cartesian product K d 1 +1 × K d 2 +1 is the graph of the Cartesian product T ( d 1 ) × T ( d 2 ) of a d 1 -dimensional simplex T ( d 1 ) and a d 2 -dimensional simplex T ( d 2 ) . And the polytope T ( d 1 ) × T ( d 2 ) is a simple polytope , a ( d 1 + d 2 ) -dimensional polytope in which every vertex is incident to exactly d 1 + d 2 edges. While not every d -polytope is ⌊ d /2⌋ -linked, it may be conjectured that every simple d -polytope is. Our result implies the veracity of the revised conjecture for Cartesian products of two simplices.

Decomposition of cartesian product of complete graphs into sunlet graphs of order eight

For any integer $k\geq 3$, we define the sunlet graph of order $2k$, denoted by $L_{2k}$, as the graph consisting of a cycle of length $k$ together with $k$ pendant vertices such that, each pendant vertex adjacent to exactly one vertex of the cycle so that the degree of each vertex in the cycle is $3$. In this paper, we establish necessary and sufficient conditions for the existence of decomposition of the Cartesian product of complete graphs into sunlet graphs of order eight.

Cartesian Products of Some Regular Graphs Admitting Antimagic Labeling for Arbitrary Sets of Real Numbers

An edge labeling of graph G with labels in A is an injection from E G to A , where E G is the edge set of G , and A is a subset of ℝ . A graph G is called ℝ -antimagic if for each subset A of ℝ with A = E G , there is an edge labeling with labels in A such that the sums of the labels assigned to edges incident to distinct vertices are different. The main result of this paper is that the Cartesian products of complete graphs (except K 1 ) and cycles are ℝ -antimagic.

Decomposition of Cartesian product of complete graphs into paths and stars with four edges

Let $P_k$ and $S_k$ denote a path and a star, respectively, on $k$ vertices. We give necessary and sufficient conditions for the existence of a complete $\{P_5,S_5\}$-decomposition of Cartesian product of complete graphs.

On the security number of the Cartesian product of graphs

A secure set of a graph is, intuitively, a set that can refute any attack from the neighborhood to its subsets. Formally, it is defined as a set S⊆V(G) such that |N[X]∩S|≥|N[X]−S| for all X⊆S. Although finding a minimum secure set is a computationally intractable problem, the minimum size of secure sets, called the security number, is studied for some specific graphs. Especially, determining the security number of the Cartesian product of graphs is one of the developed directions in this area. In this paper, we present an upper bound on the security number of the Cartesian product of general graphs, which is tight for some sparse graphs. We then determine the security number of K3m□K3n, the Cartesian product of complete graphs K3m and K3n, as well as good lower and upper bounds on the security number of the Cartesian product of complete graphs with any number of vertices.

The Radio Number for Some Classes of the Cartesian Products of Complete Graphs and Cycles

A radio coloring of graphs is a modification of the frequency assignment problem. For a connected simple graph G, a mapping g of the vertices of G to the positive integers (colors) such that for every pair u and v of G, | g(u) − g(v)| is at least 1 + diam(G) − d(u, v), is called a radio coloring of G. The largest color used by g is called span of g, denoted by rn(g). The radio number, rn(G), is the least of { rn(g) : g is a radio coloring of G }. In this paper, for n ⩾ 7 we obtain the radio number of Cartesian product of complete graph K n and cycle C m , K n ☐C m , for n even and m odd, and for n odd and m 5 (mod 8).

Graphs over Graded Rings and Relation with Hamming Graph

Hamming graph is known to be an important class of graphs, and it is a challenge to obtain algorithms that recognize whether a given graph G is a Hamming graph. Let G be a group and $$S\subseteq G$$ be a nonempty subset of G. The Cayley graph with respect to S is a graph whose vertex set is G and arcset is the set of pairs (u, v) such that $$v=su$$ for some element $$s\in S$$ . This graph is denoted by $$\mathrm{Cay}(G,S)$$ .Let $$R=\oplus _{i}R_{i}$$ be a graded ring, S be the set of homogeneous elements of R, $$S'$$ a subset of S, and $$S''=\oplus _{i\ge k}R_{i}$$ . In this paper, with a different view, we study $$\mathrm{Cay}(R, S')$$ as a generalization of $$\mathrm{Cay}(R, S)$$ to obtain a new point of view to study Cartesian products of complete graphs (Hamming graph). In particular, we show that any Hamming graph over sets of prime power sizes is isomorphic to $$\mathrm{Cay}(R,S')$$ for some graded ring R and a subset $$S'\subseteq S$$ . Also we study $$\mathrm{Cay}(R, S'')$$ as another Cayley graph over graded rings and obtain relations between this graph and total, cototal and counit graphs.

Wiener index via wirelength of an embedding

Embeddings are often viewed as a high-level representation of systematic methods to simulate an algorithm designed for one kind of parallel machine on a different network structure and/or techniques to distribute data/program variables to achieve optimum use of all available processors. A topological index is a numeric quantity of a molecule that is mathematically derived in an unambiguous way from the structural graph of a molecule. In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. Arguably, the best known of these indices is the Wiener index, defined as the sum of all distances between distinct vertices. In this paper, we have obtained the exact wirelength of embedding Cartesian products of complete graphs into a Cartesian product of paths and cycles, and generalized book. In addition to that, we have found the Wiener index of generalized book and the relation between the Wiener index and wirelength of an embedding, which solves (partially) an open problem proposed in Kumar et al. [K. J. Kumar, S. Klavžar, R. S. Rajan, I. Rajasingh and T. M. Rajalaxmi, An asymptotic relation between the wirelength of an embedding and the Wiener index, submitted to the journal].

Various characterizations of throttling numbers

Zero forcing can be described as a graph process that uses a color change rule in which vertices change white vertices to blue. The throttling number of a graph minimizes the sum of the number of vertices initially colored blue and the number of time steps required to color the entire graph. Positive semidefinite (PSD) zero forcing is a commonly studied variant of standard zero forcing that alters the color change rule. This paper introduces a method for extending a graph using a PSD zero forcing process. Using this extension method, graphs with PSD throttling number at most t are characterized as specific minors of the Cartesian product of complete graphs and trees. A similar characterization is obtained for the minor monotone floor of PSD zero forcing. Finally, the set of connected graphs on n vertices with throttling number at least n−k is characterized by forbidding a finite family of induced subgraphs. These forbidden subgraphs are constructed for standard throttling.

Phase transition for the interchange and quantum Heisenberg models on the Hamming graph

We study a family of random permutation models on the Hamming graph $H(2,n)$ (i.e., the $2$-fold Cartesian product of complete graphs), containing the interchange process and the cycle-weighted interchange process with parameter $\theta > 0$. This family contains the random walk representation of the quantum Heisenberg ferromagnet. We show that in these models the cycle structure of permutations undergoes a \textit{phase transition} -- when the number of transpositions defining the permutation is $\leq c n^2$, for small enough $c > 0$, all cycles are microscopic, while for more than $\geq C n^2$ transpositions, for large enough $C > 0$, macroscopic cycles emerge with high probability. We provide bounds on values $C,c$ depending on the parameter $\theta$ of the model, in particular for the interchange process we pinpoint exactly the critical time of the phase transition. Our results imply also the existence of a phase transition in the quantum Heisenberg ferromagnet on $H(2,n)$, namely for low enough temperatures spontaneous magnetization occurs, while it is not the case for high temperatures. At the core of our approach is a novel application of the cyclic random walk, which might be of independent interest. By analyzing explorations of the cyclic random walk, we show that sufficiently long cycles of a random permutation are uniformly spread on the graph, which makes it possible to compare our models to the mean-field case, i.e., the interchange process on the complete graph, extending the approach used earlier by Schramm.

Some Results on Palette Index of Cartesian Product Graphs

Given a proper edge coloring α of a graph G, we define the palette SG(ν, α) of a vertex ν ∈ V (G) as the set of all colors appearing on edges incident to ν. The palette index š(G) of G is the minimum number of distinct palettes occurring in a proper edge coloring of G. The windmill graph Wd(n, k) is an undirected graph constructed for k ≥ 2 and n ≥ 2 by joining n copies of the complete graph Kk at a shared universal vertex. In this paper, we determine the bound on the palette index of Cartesian products of complete graphs and simple paths. We also consider the problem of determining the palette index of windmill graphs. In particular, we show that for any positive integers n, k ≥ 2, š(Wd(n, 2k)) = n + 1.

On dynamic colouring of cartesian product of complete graph with some graphs

ABSTRACT A proper vertex colouring is called a 2-dynamic colouring, if for every vertex v with degree at least 2, the neighbours of v receive at least two colours. The smallest integer k such that G has a dynamic colouring with k colours denoted by . We denote the cartesian product of G and H by . In this paper, we find the 2-dynamic chromatic number of cartesian product of complete graph with complete graph , complete graph with complete bipartite graph and wheel graph with complete graph .

Proper Coloring Distance in Edge-Colored Cartesian Products of Complete Graphs and Cycles

An path that is edge-colored is called proper if no two consecutive edges receive the same color. A general graph that is edge-colored is called properly connected if, for every pair of vertices in the graph, there exists a properly colored path from one to the other. Given two vertices u and v in a properly connected graph G, the proper distance is the length of the shortest properly colored path from u to v. By considering a specific class of colorings that are properly connected for Cartesian products of complete and cyclic graphs, we present results on the proper distance between all pairs of vertices in the graph.

On the skewness of Cartesian products with trees

The skewness of a graph G is the minimum number of edges in G whose removal results in a planar graph. It is a parameter that measures how non-planar a graph is, and it also has important applications to VLSI design, but there are few results for skewness of graphs. In this paper, we first prove that the skewness is additive for the Zip product under certain conditions. We then present results on the lower bounds for the skewness of Cartesian products of graphs with trees and paths, respectively. Some exact values of the skewness for Cartesian products of complete graphs with trees, as well as of stars and wheels with paths are obtained by applying these lower bounds.