Direct Product Of Graphs Research Articles

The Super Restricted Edge-connectedness of Direct Product Graphs

Let [Formula: see text] be a graph with vertex set [Formula: see text] and edge set [Formula: see text]. An edge subset [Formula: see text] is called a restricted edge-cut if [Formula: see text] is disconnected and has no isolated vertices. The restricted edge-connectivity [Formula: see text] of [Formula: see text] is the cardinality of a minimum restricted edge-cut of [Formula: see text] if it has any; otherwise [Formula: see text]. If [Formula: see text] is not a star and its order is at least four, then [Formula: see text], where [Formula: see text]. The graph [Formula: see text] is said to be maximally restricted edge-connected if [Formula: see text]; the graph [Formula: see text] is said to be super restricted edge-connected if every minimum restricted edge-cut isolates an edge from [Formula: see text]. The direct product of graphs [Formula: see text] and [Formula: see text], denoted by [Formula: see text], is the graph with vertex set [Formula: see text], where two vertices [Formula: see text] and [Formula: see text] are adjacent in [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. In this paper, we give a sufficient condition for [Formula: see text] to be super restricted edge-connected, where [Formula: see text] is the complete graph on [Formula: see text] vertices.

On Independent Domination in Direct Products

In [16] Nowakowski and Rall listed a series of conjectures involving several different graph products. In particular, they conjectured that \(i(G\times H) \ge i(G)i(H)\) where i(G) is the independent domination number of G and \(G\times H\) is the direct product of graphs G and H. We show this conjecture is false, and, in fact, construct pairs of graphs for which \(\min \{i(G), i(H)\} - i(G\times H)\) is arbitrarily large. We also give the exact value of \(i(G\times K_n)\) when G is either a path or a cycle.

On well-dominated direct, Cartesian and strong product graphs

If each minimal dominating set in a graph is a minimum dominating set, then the graph is called well-dominated. Since the seminal paper on well-dominated graphs appeared in 1988, the structure of well-dominated graphs from several restricted classes has been studied. In this paper we give a complete characterization of nontrivial direct products that are well-dominated. We prove that if a strong product is well-dominated, then both of its factors are well-dominated. When one of the factors of a strong product is a complete graph, the other factor being well-dominated is also a sufficient condition for the product to be well-dominated. Our main result gives a complete characterization of well-dominated Cartesian products in which at least one of the factors is a complete graph. In addition, we conjecture that this result is actually a complete characterization of the class of nontrivial, well-dominated Cartesian products.

Strong cospectrality and twin vertices in weighted graphs

We explore algebraic and spectral properties of weighted graphs containing twin vertices that are useful in quantum state transfer. We extend the notion of adjacency strong cospectrality to Hermitian matrices, with focus on the generalized adjacency matrix and the generalized normalized adjacency matrix. We then determine necessary and sufficient conditions such that a pair of twin vertices in a weighted graph exhibits strong cospectrality with respect to the above-mentioned matrices. We also determine when strong cospectrality is preserved under Cartesian and direct products of graphs. Moreover, we generalize known results about equitable and almost equitable partitions and use these to determine which joins of the form $X\vee H$, where $X$ is either the complete or empty graph, exhibit strong cospectrality.

Super Connected Direct Product of Graphs and Cycles

The topology of an interconnection network can be modeled by a graph G=(V(G),E(G)). The connectivity of graph G is a parameter used to measure the reliability of a corresponding network. The direct product is an important graph product. This paper mainly focuses on the super connectedness of the direct product of graphs and cycles. The connectivity of G, denoted by κ(G), is the size of a minimum vertex set S⊆V(G) such that G−S is not connected or has only one vertex. The graph G is said to be super connected, simply super-κ, if every minimum vertex cut is the neighborhood of a vertex with minimum degree. The direct product of two graphs G and H, denoted by G×H, is the graph with vertex set V(G×H)=V(G)×V(H) and edge set E(G×H)={(u1,v1)(u2,v2)|u1u2∈E(G),v1v2∈E(H)}. In this paper, we give some sufficient conditions for the direct product G×Cn to be super connected, where Cn is the cycle on n vertices. Furthermore, those sufficient conditions are the best possible.

Super connectivity of a family of direct product graphs

Let and be two graphs. The Kronecker product has vertex set and the edge set In this paper we show that if is a complete multipartite graph, where the parameters satisfying certain conditions and is a path of length n−1, then is not super i-connected, where and . Also we show that is not super connected, where is a cycle of length n and .

Tightness of domination inequalities for direct product graphs

A set D of vertices in a graph G is called dominating if every vertex of G is either in D or adjacent to a vertex of D. The domination number γ(G) is the minimum size of a dominating set in G, the paired domination numberγpr(G) is the minimum size of a dominating set whose induced subgraph admits a perfect matching, and the upper domination numberΓ(G) is the maximum size of a minimal dominating set. In this paper, we investigate the sharpness of multiplicative inequalities involving the domination number and these variants, where the graph product is the direct product ×.We show that for every positive constant c, there exist graphs G and H of arbitrarily large diameter such that γ(G×H)≤cγ(G)γ(H), thus answering two questions of Paulraja and Sampath Kumar involving the paired domination number. We then study when the inequality γpr(G×H)≤12γpr(G)γpr(H) is satisfied, in particular proving that it holds whenever G and H are trees. Finally, we demonstrate that the bound Γ(G×H)≥Γ(G)Γ(H), due to Brešar, Klavžar, and Rall, is tight.

Clique Dominating Sets of Direct Product Graph of Cayley Graphs with Arithmetic Graphs

Graph Theory is one of the most flourishing branches of modern Mathematics finding widest applications in all most all branches of Science & Technology. It is applied in diverse areas such as social sciences, linguistics, physical sciences, communication engineering etc. Number Theory is one of the oldest branches of Mathematics, which inherited rich contributions from almost all greatest mathematicians, ancient and modern. Every branch of mathematics employs some notion of a product that enables the combination or decomposition of its elemental structures. Product of graphs are introduced in Graph Theory very recently and developing rapidly. In this paper, we consider direct product graphs of Cayley graphs with Arithmetic graphs and present Matching dominating set of these graphs.

Roman domination in direct product graphs and rooted product graphs

<abstract><p>Let $ G $ be a graph with vertex set $ V(G) $. A function $ f:V(G)\rightarrow \{0, 1, 2\} $ is a Roman dominating function on $ G $ if every vertex $ v\in V(G) $ for which $ f(v) = 0 $ is adjacent to at least one vertex $ u\in V(G) $ such that $ f(u) = 2 $. The Roman domination number of $ G $ is the minimum weight $ \omega(f) = \sum_{x\in V(G)}f(x) $ among all Roman dominating functions $ f $ on $ G $. In this article we study the Roman domination number of direct product graphs and rooted product graphs. Specifically, we give several tight lower and upper bounds for the Roman domination number of direct product graphs involving some parameters of the factors, which include the domination, (total) Roman domination, and packing numbers among others. On the other hand, we prove that the Roman domination number of rooted product graphs can attain only three possible values, which depend on the order, the domination number, and the Roman domination number of the factors in the product. In addition, theoretical characterizations of the classes of rooted product graphs achieving each of these three possible values are given.</p></abstract>

Total connected domination game

The (total) connected domination game on a graph \(G\) is played by two players, Dominator and Staller, according to the standard (total) domination game with the additional requirement that at each stage of the game the selected vertices induce a connected subgraph of \(G\). If Dominator starts the game and both players play optimally, then the number of vertices selected during the game is the (total) connected game domination number (\(\gamma_{\rm tcg}(G)\)) \(\gamma_{\rm cg}(G)\) of \(G\). We show that \(\gamma_{\rm tcg}(G) \in \{\gamma_{\rm cg}(G),\gamma_{\rm cg}(G) + 1,\gamma_{\rm cg}(G) + 2\}\), and consequently define \(G\) as Class \(i\) if \(\gamma_{\rm tcg}(G) = \gamma_{\rm cg} + i\) for \(i \in \{0,1,2\}\). A large family of Class \(0\) graphs is constructed which contains all connected Cartesian product graphs and connected direct product graphs with minumum degree at least \(2\). We show that no tree is Class \(2\) and characterize Class \(1\) trees. We provide an infinite family of Class \(2\) bipartite graphs.

Dominating the Direct Product of Two Graphs through Total Roman Strategies

Given a graph G without isolated vertices, a total Roman dominating function for G is a function f:V(G)→{0,1,2} such that every vertex u with f(u)=0 is adjacent to a vertex v with f(v)=2, and the set of vertices with positive labels induces a graph of minimum degree at least one. The total Roman domination number γtR(G) of G is the smallest possible value of ∑v∈V(G)f(v) among all total Roman dominating functions f. The total Roman domination number of the direct product G×H of the graphs G and H is studied in this work. Specifically, several relationships, in the shape of upper and lower bounds, between γtR(G×H) and some classical domination parameters for the factors are given. Characterizations of the direct product graphs G×H achieving small values (≤7) for γtR(G×H) are presented, and exact values for γtR(G×H) are deduced, while considering various specific direct product classes.

Total colouring of some cartesian and direct product graphs

A graph is $k$-total colourable if there is an assignment of $k$ different colours to the vertices and edges of the graph such that no two adjacent nor incident elements receive the same colour. The total chromatic number of some direct product graphs are determined. In particular, a sufficient condition is given for direct products of bipartite graphs to have total chromatic number equal to its maximum degree plus one. Partial results towards the total chromatic number of $K_n\times K_m$ are also established.

Enumerating cliques in direct product graphs

The unitary Cayley graph of $\mathbb Z/n\mathbb Z$, denoted $G_{\mathbb Z/n\mathbb Z}$, is the graph with vertices $0,1,\ldots,$ $n-1$ in which two vertices are adjacent if and only if their difference is relatively prime to $n$. These graphs are central to the study of graph representations modulo integers, which were originally introduced by Erdős and Evans. We give a brief account of some results concerning these beautiful graphs and provide a short proof of a simple formula for the number of cliques of any order $m$ in the unitary Cayley graph $G_{\mathbb Z/n\mathbb Z}$. This formula involves an exciting class of arithmetic functions known as Schemmel totient functions, which we also briefly discuss. More generally, the proof yields a formula for the number of cliques of order $m$ in a direct product of balanced complete multipartite graphs.

Asymmetric colorings of products of graphs and digraphs

We extend results about asymmetric colorings of finite Cartesian products of graphs to strong and direct products of graphs and digraphs. On the way we shorten proofs for the existence of prime factorizations of finite digraphs and characterize the structure of the automorphism groups of strong and direct products. The paper ends with results on asymmetric colorings of Cartesian products of finite and infinite digraphs.

A note on the connectivity of direct products of graphs

In recent years, the vertex connectivity and edge connectivity have aroused great interest of many scholars in the field of mathematics. We consider in this note that edge connectivity of a graph called k ′(G). In this note, If G is a bipartite graph with vertices n ≥ 3 we prove that k ′(G × Kn) = min {n(n – 1) k ′(G), (n – 1) δ(G)}, and their edge connectivity is satisfied k ′(G) = δ(G), we can obtain k ′(G × Kn) = δ (G × Kn) = (n – 1) δ(G).

Domination and upper domination of direct product graphs

Let XZ∕nZ denote the unitary Cayley graph of Z∕nZ. We present results on the tightness of the known inequality γ(XZ∕nZ)≤γt(XZ∕nZ)≤g(n), where γ andγt denote the domination number and total domination number, respectively, and g is the arithmetic function known as Jacobsthal’s function. In particular, we construct integers n with arbitrarily many distinct prime factors such that γ(XZ∕nZ)≤γt(XZ∕nZ)≤g(n)−1. We give lower bounds for the domination numbers of direct products of complete graphs and present a conjecture for the exact values of the upper domination numbers of direct products of balanced, complete multipartite graphs.

Hyperbolicity of Direct Products of Graphs

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G1×G2 is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs).

Neighborhood Reconstruction and Cancellation of Graphs

We connect two seemingly unrelated problems in graph theory.Any graph $G$ has a neighborhood multiset $\mathscr{N}(G)= \{N(x) \mid x\in V(G)\}$ whose elements are precisely the open vertex-neighborhoods of $G$. In general there exist non-isomorphic graphs $G$ and $H$ for which $\mathscr{N}(G)=\mathscr{N}(H)$. The neighborhood reconstruction problem asks the conditions under which $G$ is uniquely reconstructible from its neighborhood multiset, that is, the conditions under which $\mathscr{N}(G)=\mathscr{N}(H)$ implies $G\cong H$. Such a graph is said to be neighborhood-reconstructible.The cancellation problem for the direct product of graphs seeks the conditions under which $G\times K\cong H\times K$ implies $G\cong H$. Lovász proved that this is indeed the case if $K$ is not bipartite. A second instance of the cancellation problem asks for conditions on $G$ that assure $G\times K\cong H\times K$ implies $G\cong H$ for any bipartite~$K$ with $E(K)\neq \emptyset$. A graph $G$ for which this is true is called a cancellation graph.We prove that the neighborhood-reconstructible graphs are precisely the cancellation graphs. We also present some new results on cancellation graphs, which have corresponding implications for neighborhood reconstruction. We are particularly interested in the (yet-unsolved) problem of finding a simple structural characterization of cancellation graphs (equivalently, neighborhood-reconstructible graphs).

Game Chromatic Number of Direct Product of Some Families of Graphs

In this paper, we determine the exact values of the game chromatic number of direct product graphs K1,n×K1,m, Km,n×Ka,b, Pn×K1,m, P2×Wn and P2×Cn.

Associativity and Non-Associativity of Some Hypergraph Products

Several variants of hypergraph products have been introduced as generalizations of the strong and direct products of graphs. Here we show that only some of them are associative. In addition to the Cartesian product, these are the minimal rank preserving direct product, and the normal product. Counter-examples are given for the strong product as well as the non-rank-preserving and the maximal rank preserving direct product.