scholarly journals The Geometric Kernel of Integral Circulant Graphs

10.37236/9764 ◽  
2021 ◽  
Vol 28 (3) ◽  
Author(s):  
J. W. Sander
Keyword(s):  
Adjacency Matrix ◽  
Euclidean Plane ◽  
Rotational Symmetry ◽  
Central Hole ◽  
Geometric Feature ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Notable Feature ◽  
Arithmetic Properties ◽  
Vertex Set

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 $\mathrm{ICG}(n,\mathcal{D})$.

scholarly journals The energy of integral circulant graphs with prime power order

2011 ◽  
Vol 5 (1) ◽  
pp. 22-36 ◽  
Author(s):  
J.W. Sander ◽  
T. Sander
Keyword(s):  
Adjacency Matrix ◽  
Prime Power ◽  
Prime Power Order ◽  
Circulant Graph ◽  
Closed Formula ◽  
Circulant Graphs ◽  
Vertex Set

The energy of a graph is the sum of the moduli of the eigenvalues of its adjacency matrix. We study the energy of integral circulant graphs, also called gcd graphs. Such a graph can be characterized by its vertex count n and a set D of divisors of n such that its vertex set is Zn and its edge set is {{a,b} : a, b ? Zn; gcd(a-b, n)? D}. For an integral circulant graph on ps vertices, where p is a prime, we derive a closed formula for its energy in terms of n and D. Moreover, we study minimal and maximal energies for fixed ps and varying divisor sets D.

HAMILTONIAN PROPERTIES OF FAULTY RECURSIVE CIRCULANT GRAPHS

2002 ◽  
Vol 03 (03n04) ◽  
pp. 273-289 ◽  
Author(s):  
CHANG-HSIUNG TSAI ◽  
JIMMY J. M. TAN ◽  
YEN-CHU CHUANG ◽  
LIH-HSING HSU
Keyword(s):  
Bipartite Graph ◽  
Hamiltonian Path ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Vertex Set ◽  
Regular Node ◽  
Hamiltonian Properties

We present some results concerning hamiltonian properties of recursive circulant graphs in the presence of faulty vertices and/or edges. The recursive circulant graph G(N, d) with d ≥ 2 has vertex set V(G) = {0, 1, …, N - 1} and the edge set E(G) = {(v, w)| ∃ i, 0 ≤ i ≤ ⌈ log d N⌉ - 1, such that v = w + di (mod N)}. When N = cdk where d ≥ 2 and 2 ≤ c ≤ d, G(cdk, d) is regular, node symmetric and can be recursively constructed. G(cdk, d) is a bipartite graph if and only if c is even and d is odd. Let F, the faulty set, be a subset of V(G(cdk, d)) ∪ E(G(cdk, d)). In this paper, we prove that G(cdk, d) - F remains hamiltonian if |F| ≤ deg (G(cdk, d)) - 2 and G(cdk, d) is not bipartite. Moreover, if |F| ≤ deg (G(cdk, d)) - 3 and G(cdk, d) is not a bipartite graph, we prove a more stronger result that for any two vertices u and v in V(G(cdk, d)) - F, there exists a hamiltonian path of G(cdk, d) - F joining u and v.

scholarly journals Pretty Good State Transfer on Circulant Graphs

10.37236/6388 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Hiranmoy Pal ◽  
Bikash Bhattacharjya
Keyword(s):  
Complex Number ◽  
Adjacency Matrix ◽  
Transition Matrix ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Real Numbers ◽  
Cartesian Products ◽  
Edge Disjoint ◽  
State Transfer ◽  
Good State

Let $G$ be a graph with adjacency matrix $A$. The transition matrix of $G$ relative to $A$ is defined by $H(t):=\exp{\left(-itA\right)}$, where $t\in {\mathbb R}$. The graph $G$ is said to admit pretty good state transfer between a pair of vertices $u$ and $v$ if there exists a sequence of real numbers $\{t_k\}$ and a complex number $\gamma$ of unit modulus such that $\lim\limits_{k\rightarrow\infty} H(t_k) e_u=\gamma e_v.$ We find that the cycle $C_n$ as well as its complement $\overline{C}_n$ admit pretty good state transfer if and only if $n$ is a power of two, and it occurs between every pair of antipodal vertices. In addition, we look for pretty good state transfer in more general circulant graphs. We prove that union (edge disjoint) of an integral circulant graph with a cycle, each on $2^k$ $(k\geq 3)$ vertices, admits pretty good state transfer. The complement of such union also admits pretty good state transfer. Using Cartesian products, we find some non-circulant graphs admitting pretty good state transfer.

Integral circulant graphs with four distinct eigenvalues

2018 ◽  
Vol 10 (05) ◽  
pp. 1850057 ◽  
Author(s):  
T. Tamizh Chelvam ◽  
S. Raja
Keyword(s):  
Cyclic Group ◽  
Undirected Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Finite Cyclic Group ◽  
Vertex Set

Let [Formula: see text], the finite cyclic group of order [Formula: see text]. Assume that [Formula: see text] and [Formula: see text]. The circulant graph [Formula: see text] is the undirected graph having the vertex set [Formula: see text] and edge set [Formula: see text]. Let [Formula: see text] be a set of positive, proper divisors of the integer [Formula: see text]. In this paper, by using [Formula: see text] we characterize certain connected integral circulant graphs with four distinct eigenvalues.

scholarly journals Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Hamamache Kheddouci ◽  
Olivier Togni
Keyword(s):  
Distance Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Feedback Vertex Set ◽  
Distance Graphs ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
International Audience ◽  
Minimum Feedback Vertex Set ◽  
Vertex Sets

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

scholarly journals Bounds for minimum feedback vertex sets in distance graphs and circulant graphs

2008 ◽  
Vol Vol. 10 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Hamamache Kheddouci ◽  
Olivier Togni
Keyword(s):  
Distance Graph ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Feedback Vertex Set ◽  
Distance Graphs ◽  
Vertex Set ◽  
Feedback Vertex Set Problem ◽  
International Audience ◽  
Minimum Feedback Vertex Set ◽  
Vertex Sets

Graphs and Algorithms International audience For a set D ⊂ Zn, the distance graph Pn(D) has Zn as its vertex set and the edges are between vertices i and j with |i − j| ∈ D. The circulant graph Cn(D) is defined analogously by considering operations modulo n. The minimum feedback vertex set problem consists in finding the smallest number of vertices to be removed in order to cut all cycles in the graph. This paper studies the minimum feedback vertex set problem for some families of distance graphs and circulant graphs depending on the value of D.

scholarly journals Recognizing Circulant Graphs of Prime Order in Polynomial Time

10.37236/1363 ◽  
1998 ◽  
Vol 5 (1) ◽  
Author(s):  
Mikhail E. Muzychuk ◽  
Gottfried Tinhofer
Keyword(s):  
Adjacency Matrix ◽  
Prime Order ◽  
Circulant Matrix ◽  
Recognition Algorithm ◽  
Association Schemes ◽  
Necessary Condition ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Coherent Configuration ◽  
Schur Ring

A circulant graph $G$ of order $n$ is a Cayley graph over the cyclic group ${\bf Z}_n.$ Equivalently, $G$ is circulant iff its vertices can be ordered such that the corresponding adjacency matrix becomes a circulant matrix. To each circulant graph we may associate a coherent configuration ${\cal A}$ and, in particular, a Schur ring ${\cal S}$ isomorphic to ${\cal A}$. ${\cal A}$ can be associated without knowing $G$ to be circulant. If $n$ is prime, then by investigating the structure of ${\cal A}$ either we are able to find an appropriate ordering of the vertices proving that $G$ is circulant or we are able to prove that a certain necessary condition for $G$ being circulant is violated. The algorithm we propose in this paper is a recognition algorithm for cyclic association schemes. It runs in time polynomial in $n$.

scholarly journals Edge metric dimension of some classes of circulant graphs

2020 ◽  
Vol 28 (3) ◽  
pp. 15-37
Author(s):  
Muhammad Ahsan ◽  
Zohaib Zahid ◽  
Sohail Zafar
Keyword(s):  
Connected Graph ◽  
Ordered Set ◽  
Metric Dimension ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Minimum Cardinality ◽  
Vertex Set ◽  
Metric Basis

AbstractLet 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.

scholarly journals Restricted triangulation on circulant graphs

2018 ◽  
Vol 16 (1) ◽  
pp. 358-369 ◽  
Author(s):  
Niran Abbas Ali ◽  
Adem Kilicman ◽  
Hazim Michman Trao
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Existence Problem ◽  
Circulant Graph ◽  
Circulant Graphs ◽  
Value Set ◽  
Vertex Set ◽  
Necessary And Sufficient

AbstractThe restricted triangulation existence problem on a given graph decides whether there exists a triangulation on the graph’s vertex set that is restricted with respect to its edge set. Let G = C(n, S) be a circulant graph on n vertices with jump value set S. We consider the restricted triangulation existence problem for G. We determine necessary and sufficient conditions on S for which G admitting a restricted triangulation. We characterize a set of jump values S(n) that has the smallest cardinality with C(n, S(n)) admits a restricted triangulation. We present the measure of non-triangulability of Kn − G for a given G.

The Metric Dimension of Circulant Graphs

2017 ◽  
Vol 60 (1) ◽  
pp. 206-216 ◽  
Author(s):  
Tomáš Vetrík
Keyword(s):  
Metric Dimension ◽  
Circulant Graph ◽  
Resolving Set ◽  
Circulant Graphs ◽  
Vertex Set ◽  
Infinite Set

Abstract. A subsetWof the vertex set of a graphGis called aresolving setofGif for every pair of distinct verticesu,vofG, there isw∊Wsuch that the distance ofwanduis different from the distance ofwandv. The cardinality of a smallest resolving set is called the metric dimension ofG, denoted by dim(G). The circulant graphCn(1, 2, . . . ,t) consists of the verticesv0,v1, . . . ,vn−1and the edgesvivi+j, where 0 ≤i≤n− 1, 1 ≤j≤t(), the indices are taken modulon. Grigorious, Manuel, Miller, Rajan, and Stephen proved that dim(Cn(1, 2, . . . ,t)) ≥t+ 1 for, and they presented a conjecture saying that dim(Cn(1, 2, . . . ,t)) =t+p− 1 forn= 2tk+t+p, where 3 ≤p≤t+ 1. We disprove both statements. We show that ift≥ 4 is even, there exists an infinite set of values ofnsuch that dim(Cn(1, 2, . . . ,t)) =t. We also prove that dim(Cn(1, 2, . . . ,t)) ≤t+p/2 forn= 2tk+t+p, wheretandpare even,t≥ 4, 2 ≤p≤t, andk≥ 1.

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