Circulant Graphs Research Articles

Formula omitted]-packing colorings of distance graphs [formula omitted

Given a graph G and a non-decreasing sequence S=(a1,a2,…) of positive integers, the mapping f:V(G)→{1,…,k} is an S-packing k-coloring of G if for any distinct vertices u,v∈V(G) with f(u)=f(v)=i the distance between u and v in G is greater than ai. The smallest k such that G has an S-packing k-coloring is the S-packing chromatic number, χS(G), of G. In this paper, we continue the study of S-packing colorings of distance graphs, initiated by Togni (2014) and Ekstein et al. (2012). We focus on the distance graphs G(Z,{2,t}), where t>1 is an odd integer, which has Z as its vertex set, and i,j∈Z are adjacent if |i−j|∈{2,t}. We determine the S-packing chromatic numbers of the graphs G(Z,{2,t}), where S is any sequence with ai∈{1,2} for all i. In addition, we give lower and upper bounds for the d-distance chromatic numbers of the distance graphs G(Z,{2,t}), which in the cases d≥t−3 give the exact values. Implications for the corresponding S-packing chromatic numbers of the circulant graphs are also discussed.

L(3,1)-labeling of circulant graphs

An [Formula: see text]-labeling of a graph [Formula: see text] is an assignment of non-negative integers to the vertices such that if two vertices [Formula: see text] and [Formula: see text] are adjacent then they receive labels that differ by at least [Formula: see text], and when [Formula: see text] and [Formula: see text] are not adjacent but there is a two-hop path between them, then they receive labels that differ by at least one. The span [Formula: see text] of such a labeling is the difference between the largest and the smallest vertex labels assigned. Let [Formula: see text] denote the least [Formula: see text] such that [Formula: see text] admits an [Formula: see text]-labeling using labels from [Formula: see text]. A Cayley graph of group [Formula: see text] is called circulant graph of order [Formula: see text], if [Formula: see text]. In this paper initially we investigate the [Formula: see text]-labeling for circulant graphs with “large” connection sets, and then we extend our observation and find the span of [Formula: see text]-labeling for any circulants of order [Formula: see text].

Correction to: A complete classification of 5-regular circulant graphs that allow cyclic orthogonal double covers

Correction to: A complete classification of 5-regular circulant graphs that allow cyclic orthogonal double covers

The Waring’s problem over finite fields through generalized Paley graphs

We show that the Waring number over a finite field Fq, denoted as g(k,q), when exists coincides with the diameter of the generalized Paley graph Γ(k,q)=Cay(Fq,Rk) with Rk={xk:x∈Fq∗}. We find infinite new families of exact values of g(k,q) from a characterization of graphs Γ(k,q) which are also Hamming graphs proved by Lim and Praeger in 2009. Then, we show that every positive integer is the Waring number for some pair (k,q) with q not a prime. Finally, we find a lower bound for g(k,p) with p prime by using that Γ(k,p) is a circulant graph in this case.

On the partition dimension of directed circulant graphs

The directed circulant graph Cn (1, 2, … , t) consists of the vertices v 0 , v 1 , … , vn -1 and edges directed from vi to vi +j for every i = 0, 1, … , n – 1, and j = 1, 2, … , t, where 1 ≤ t ≤ n – 1, the indices are taken modulo n. We present lower and upper bounds on the partition dimension of very general graphs Cn (1, 2, … , t). Our bounds yield exact values of the partition dimension of the graphs Cn (1, 2, … , t) for t = 2, 3, 4 and n ≥ 10.

A Breadth-first Search Tree Construction for Multiplicative Circulant Graphs

In this paper, we give a recursive method in constructing a breadth-first search tree for multiplicative circulant graphs of order power of odd. We then use the proposed construction in reproving some results concerning multiplicative circulant graph's diameter, average distance and distance spectral radius. We also determine the graph's Wiener index, vertex-forwarding index, and a bound for its edge-forwarding index. Finally, we discuss some possible research works in which the proposed construction can be applied.

A complete classification of 5-regular circulant graphs that allow cyclic orthogonal double covers

Graph theory and its applications attract several researchers from different areas of research. It is used to model wide range of systems, for example in biological system and in studying chaotic systems. Decomposition of complex graphs into simple small graphs is very helpful to study that complex systems. A complete classification is given for circulant graphs of degree 5 that allow an Orthogonal Double Cover by some graphs. Whenever such a cover exists, an orthogonal labelling for the corresponding covering graph is presented. Our work here completes all previous incomplete work about this point.

Induced H-packing k-partition of graphs

ABSTRACT The minimum induced H-packing k-partition number is denoted by . The induced H-packing k-partition number denoted by is defined as where the minimum is taken over all H-packings of G. In this paper, we obtain the induced -packing k-partition number for trees, slim trees, split graphs, complete bipartite graphs, grids and circulant graphs. We also deal with networks having perfect -packing where is a claw on four vertices. We prove that an induced -packing k-partition problem is NP-Complete. Further we prove that the induced -packing k-partition number of is 2 for all hypercube networks with perfect -packing and prove that for all locally twisted cubes with perfect -packing.

Removing Symmetry in Circulant Graphs and Point-Block Incidence Graphs

An automorphism of a graph is a mapping of the vertices onto themselves such that connections between respective edges are preserved. A vertex v in a graph G is fixed if it is mapped to itself under every automorphism of G. The fixing number of a graph G is the minimum number of vertices, when fixed, fixes all of the vertices in G. The determination of fixing numbers is important as it can be useful in determining the group of automorphisms of a graph-a famous and difficult problem. Fixing numbers were introduced and initially studied by Gibbons and Laison, Erwin and Harary and Boutin. In this paper, we investigate fixing numbers for graphs with an underlying cyclic structure, which provides an inherent presence of symmetry. We first determine fixing numbers for circulant graphs, showing in many cases the fixing number is 2. However, we also show that circulant graphs with twins, which are pairs of vertices with the same neighbourhoods, have considerably higher fixing numbers. This is the first paper that investigates fixing numbers of point-block incidence graphs, which lie at the intersection of graph theory and combinatorial design theory. We also present a surprising result-identifying infinite families of graphs in which fixing any vertex fixes every vertex, thus removing all symmetries from the graph.

Asymptotic energy of connected cubic circulant graphs

In this article, we compute the oblique asymptote of the energy function for all connected cubic circulant graphs. Moreover, we show that this oblique asymptote is an upper bound for the energies of two of the subclasses of Möbius ladder graphs and lower bound for the remaining four subclasses.

On 2-partition dimension of the circulant graphs

The partition dimension is a variant of metric dimension in graphs. It has arising applications in the fields of network designing, robot navigation, pattern recognition and image processing. Let G (V (G) , E (G)) be a connected graph and Γ = {P1, P2, …, Pm} be an ordered m-partition of V (G). The partition representation of vertex v with respect to Γ is an m-vector r (v|Γ) = (d (v, P1) , d (v, P2) , …, d (v, Pm)), where d (v, P) = min {d (v, x) |x ∈ P} is the distance between v and P. If the m-vectors r (v|Γ) differ in at least 2 positions for all v ∈ V (G), then the m-partition is called a 2-partition generator of G. A 2-partition generator of G with minimum cardinality is called a 2-partition basis of G and its cardinality is known as the 2-partition dimension of G. Circulant graphs outperform other network topologies due to their low message delay, high connectivity and survivability, therefore are widely used in telecommunication networks, computer networks, parallel processing systems and social networks. In this paper, we computed partition dimension of circulant graphs Cn (1, 2) for n ≡ 2 (mod 4), n ≥ 18 and hence corrected the result given by Salman et al. [Acta Math. Sin. Engl. Ser. 2012, 28, 1851-1864]. We further computed the 2-partition dimension of Cn (1, 2) for n ≥ 6.

Irregularity Strength of Circulant Graphs Using Algorithmic Approach

This paper deals with decomposition of complete graphs on $n$ vertices into circulant graphs with reduced degree $r . They are denoted as $C_{n}(a_{1}, a_{2}, {\dots }, a_{m})$ , where $a_{1}$ to $a_{m}$ are generators. Mathematical labeling for such bigger (higher order and huge size) and complex (strictly regular with so many triangles) graphs is very difficult. That is why after decomposition, an edge irregular $k$ -labeling for these subgraphs is computed with the help of algorithmic approach. Results of $k$ are computed by implementing this iterative algorithm in computer. Using the values of $k$ , an upper bound for edge irregularity strength is suggested for $C_{n}(a_{1}, a_{2}, {\dots }, a_{m})$ that is ${\vert E\vert }/{2}\log _{2} \vert V\vert $ .

On total coloring of 4-regular circulant graphs

A k-total coloring of a graph G is an assignment of k colors to the vertices and edges (elements) of G so that adjacent or incident elements have different colors. The total chromatic number of G is the smallest integer k for which G has a k-total coloring. The well known Total Coloring Conjecture states that the total chromatic number of a graph is either ∆(G) + 1 or ∆(G) + 2, where ∆(G) is the maximum degree of G. Graphs with χ" (G) = ∆(G) + 1 are known as Type 1, and graphs with χ" (G) = ∆(G) + 2 are known as Type 2.In this work, we investigate the total coloring of circulant graphs. We establish that all members of three infinite families of 4-regular circulant graphs are Type 1, except for one graph which is Type 2. These results contribute to the conjecture proposed by Khennoufa and Togni in 2008, which states that, except for a finite number of Type 2, 4-regular circulant graphs are all Type 1.

Embedding of Circulant Networks into Wheels

Circulant graphs have been studied over many years because of its vast applications in telecommunication industry. We find the minimum wirelengths of embedding the circulant graph G(n; ±{1, 2}) and G(n; ±{1, 2, …, j}) into variation of wheel graphs on n vertices in this paper.

On the reflexive edge strength of the circulant graphs

<abstract> A labeling of a graph is an assignment that carries some sets of graph elements into numbers (usually the non negative integers). The total $ k $-labeling is an assignment $ f_{e} $ from the edge set to the set $ \{1, 2, ..., k_{e} \} $ and assignment $ f_{v} $ from the vertex set to the set $ \{0, 2, 4, ..., 2k_{v} \} $, where $ k = max\{k_{e}, 2k_{v} \} $. An edge irregular reflexive $ k $-labeling of the graph $ G $ is the total $ k $-labeling, if distinct edges have distinct weights, where the edge weight is defined as the sum of label of that edge and the labels of the end vertices. The minimum $ k $ for which the graph $ G $ has an edge irregular reflexive $ k $-labeling is called the reflexive edge strength of the graph $ G $, denoted by $ res(G) $. In this paper we study the edge reflexive irregular $ k $-labeling for some cases of circulant graphs and determine the exact value of the reflexive edge strength for several classes of circulant graphs. </abstract>

Constructing Independent Spanning Trees on Generalized Recursive Circulant Graphs

The generalized recursive circulant networking can be widely used in the design and implementation of interconnection networks. It consists of a series of processors, each is connected through bidirectional, point-to-point communication channels to different neighbors. In this work, we apply the shortest path routing concept to build independent spanning trees on the generalized recursive circulant graphs. The proposed strategy loosen the restricted conditions in previous research and extended the result to a more general vertex setting by design the specific algorithm to deal with the constraint issue.

On the balanceability of some graph classes

Given a graph G, a 2-coloring of the edges of Kn is said to contain a balanced copy of G if we can find a copy of G such that half of its edges are in each color class. If, for every sufficiently large n, there exists an integer k such that every 2-coloring of Kn with more than k edges in each color class contains a balanced copy of G, then we say that G is balanceable. Balanceability was introduced by Caro, Hansberg and Montejano, who also gave a structural characterization of balanceable graphs.In this paper, we extend the study of balanceability by finding new sufficient conditions for a graph to be balanceable or not. We use those conditions to fully characterize the balanceability of graph classes such as rectangular and triangular grids, as well as a special class of circulant graphs.

Total colorings of circulant graphs

The total chromatic number [Formula: see text] is the least number of colors needed to color the vertices and edges of a graph [Formula: see text] such that no incident or adjacent elements (vertices or edges) receive the same color. Behzad and Vizing proposed a well-known total coloring conjecture (TCC): [Formula: see text], where [Formula: see text] is the maximum degree of [Formula: see text]. For the powers of cycles, Campos and de Mello proposed the following conjecture: Let [Formula: see text] denote the graphs of powers of cycles of order [Formula: see text] and length [Formula: see text] with [Formula: see text]. Then, [Formula: see text] In this paper, we prove the Campos and de Mello’s conjecture for some classes of powers of cycles. Also, we prove the TCC for complement of powers of cycles.

Maximum nullity and zero forcing of circulant graphs

AbstractThe zero forcing number of a graph has been applied to communication complexity, electrical power grid monitoring, and some inverse eigenvalue problems. It is well-known that the zero forcing number of a graph provides a lower bound on the minimum rank of a graph. In this paper we bound and characterize the zero forcing number of various circulant graphs, including families of bipartite circulants, as well as all cubic circulants. We extend the definition of the Möbius ladder to a type of torus product to obtain bounds on the minimum rank and the maximum nullity on these products. We obtain equality for torus products by employing orthogonal Hankel matrices. In fact, in every circulant graph for which we have determined these numbers, the maximum nullity equals the zero forcing number. It is an open question whether this holds for all circulant graphs.

Edge metric dimension of some classes of circulant graphs

Abstract Let G = (V (G), E(G)) be a connected graph and x, y ∈ V (G), d(x, y) = min{ length of x − y path } and for e ∈ E(G), d(x, e) = min{d(x, a), d(x, b)}, where e = ab. A vertex x distinguishes two edges e 1 and e 2, if d(e 1, x) ≠ d(e 2, x). Let WE = {w 1, w 2, . . ., wk } be an ordered set in V (G) and let e ∈ E(G). The representation r(e | WE ) of e with respect to WE is the k-tuple (d(e, w 1), d(e, w 2), . . ., d(e, wk )). If distinct edges of G have distinct representation with respect to WE , then WE is called an edge metric generator for G. An edge metric generator of minimum cardinality is an edge metric basis for G, and its cardinality is called edge metric dimension of G, denoted by edim(G). The circulant graph Cn (1, m) has vertex set {v 1, v 2, . . ., vn } and edge set {vivi +1 : 1 ≤ i ≤ n−1}∪{vnv 1}∪{vivi+m : 1 ≤ i ≤ n−m}∪{vn−m+ivi : 1 ≤ i ≤ m}. In this paper, it is shown that the edge metric dimension of circulant graphs Cn (1, 2) and Cn (1, 3) is constant.