On Automorphisms of the Double Cover of a Circulant Graph

A graph $X$ is said to be unstable if the direct product $X \times K_2$ (also called the canonical double cover of $X$) has automorphisms that do not come from automorphisms of its factors $X$ and $K_2$. It is nontrivially unstable if it is unstable, connected, and nonbipartite, and no two distinct vertices of $X$ have exactly the same neighbors. We find three new conditions that each imply a circulant graph is unstable. (These yield infinite families of nontrivially unstable circulant graphs that were not previously known.) We also find all of the nontrivially unstable circulant graphs of order $2p$, where $p$ is any prime number. Our results imply that there does not exist a nontrivially unstable circulant graph of order $n$ if and only if either $n$ is odd, or $n < 8$, or $n = 2p$, for some prime number $p$ that is congruent to $3$ modulo $4$.

A Combinatorial Approach to the Computation of the Fractional Edge Dimension of Graphs

E. Yi recently introduced the fractional edge dimension of graphs. It has many applications in different areas of computer science such as in sensor networking, intelligent systems, optimization, and robot navigation. In this paper, the fractional edge dimension of vertex and edge transitive graphs is calculated. The class of hypercube graph Qn with an odd number of vertices n is discussed. We propose the combinatorial criterion for the calculation of the fractional edge dimension of a graph, and hence applied it to calculate the fractional edge dimension of the friendship graph Fk and the class of circulant graph Cn(1,2).

The Geometric Kernel of Integral Circulant Graphs

By a suitable representation in the Euclidean plane, each circulant graph $G$, i.e. a graph with a circulant adjacency matrix ${\mathcal A}(G)$, reveals its rotational symmetry and, as the drawing's most notable feature, a central hole, the so-called \emph{geometric kernel} of $G$. Every integral circulant graph $G$ on $n$ vertices, i.e. satisfying the additional property that all of the eigenvalues of ${\mathcal A}(G)$ are integral, is isomorphic to some graph $\mathrm{ICG}(n,\mathcal{D})$ having vertex set $\mathbb{Z}/n\mathbb{Z}$ and edge set $\{\{a,b\}:\, a,b\in\mathbb{Z}/n\mathbb{Z} ,\, \gcd(a-b,n)\in \mathcal{D}\}$ for a uniquely determined set $\mathcal{D}$ of positive divisors of $n$. A lot of recent research has revolved around the interrelation between graph-theoretical, algebraic and arithmetic properties of such graphs. In this article we examine arithmetic implications imposed on $n$ by a geometric feature, namely the size of the geometric kernel of $\mathrm{ICG}(n,\mathcal{D})$.

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].

Edge metric dimension of some classes of circulant graphs

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 e1 and e2, if d(e1, x) ≠ d(e2, x). Let WE = {w1, w2, . . ., 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, w1), d(e, w2), . . ., 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 {v1, v2, . . ., vn} and edge set {vivi+1 : 1 ≤ i ≤ n−1}∪{vnv1}∪{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.

Resolvability in Subdivision of Circulant Networks Cn1,k

Circulant networks form a very important and widely explored class of graphs due to their interesting and wide-range applications in networking, facility location problems, and their symmetric properties. A resolving set is a subset of vertices of a connected graph such that each vertex of the graph is determined uniquely by its distances to that set. A resolving set of the graph that has the minimum cardinality is called the basis of the graph, and the number of elements in the basis is called the metric dimension of the graph. In this paper, the metric dimension is computed for the graph Gn1,k constructed from the circulant graph Cn1,k by subdividing its edges. We have shown that, for k=2, Gn1,k has an unbounded metric dimension, and for k=3 and 4, Gn1,k has a bounded metric dimension.

The integer {k}-domination number of circulant graphs

Let [Formula: see text] be a simple undirected graph. [Formula: see text] is a circulant graph defined on [Formula: see text] with difference set [Formula: see text] provided two vertices [Formula: see text] and [Formula: see text] in [Formula: see text] are adjacent if and only if [Formula: see text]. For convenience, we use [Formula: see text] to denote such a circulant graph. A function [Formula: see text] is an integer [Formula: see text]-domination function if for each [Formula: see text], [Formula: see text] By considering all [Formula: see text]-domination functions [Formula: see text], the minimum value of [Formula: see text] is the [Formula: see text]-domination number of [Formula: see text], denoted by [Formula: see text]. In this paper, we prove that if [Formula: see text], [Formula: see text], then the integer [Formula: see text]-domination number of [Formula: see text] is [Formula: see text].

On Rationality of Generating Function for the Number of Spanning Trees in Circulant Graphs

Let [Formula: see text] be the generating function for the number [Formula: see text] of spanning trees in the circulant graph Cn(s1, s2, …, sk). We show that F(x) is a rational function with integer coefficients satisfying the property F(x) = F(1/x). A similar result is also true for the circulant graphs C2n(s1, s2, …, sk, n) of odd valency. We illustrate the obtained results by a series of examples.