Burning Circulant Graphs

In this paper we study the graph parameter of burning number, introduced by Bonato, Janssen, and Roshanbin (2014). We are particular interested in determining the burning number of Circulant graphs. In this paper, we find upper and lower bounds on the burning number of classes of circulant graphs of degree at most four. The burning number is found exactly for a family of circulant graphs of degree three, and two specific families of circulant graphs of degree four. Finally, we given upper and lower bounds on the burning number of families of circulant graphs of higher degree.Comment: 15 page

On congruence in Z^n and the dimension of a multidimensional circulant

From a generalization to $Z^n$ of the concept of congruence we define a family of regular digraphs or graphs called multidimensional circulants, which turn out to be Cayley (di)graphs of Abelian groups. This paper is mainly devoted to show the relationship between the Smith normal form for integral matrices and the dimensions of such (di)graphs, that is the minimum ranks of the groups they can arise from. In particular, those 2-step multidimensional circulants which are circulants, that is Cayley (di)graphs of cyclic groups, are fully characterized. In addition, a reasoning due to Lawrence is used to prove that the cartesian product of $n$ circulants with equal number of vertices $p>2$, $p$ a prime, has dimension $n$

A few c2c_2 invariants of circulant graphs

The $c_2$ invariant is an arithmetic graph invariant introduced by Schnetz and developed by Brown and Schnetz in order to better understand Feynman integrals. This document looks at the special case where the graph in question is a 4-regular circulant graph with one vertex removed; call such a graph a decompletion of a circulant graph. The $c_2$ invariant for the prime $2$ is computed in the case of the decompletion of circulant graphs $C_n(1,3)$ and $C_{2k+2}(1,k)$. For any prime $p$ and for the previous two families of circulant graphs along with the further families $C_n(1,4)$, $C_n(1,5)$, $C_n(1,6)$, $C_n(2,3)$, $C_n(2,4)$, $C_n(2,5)$, and $C_n(3,4)$, the same technique gives the $c_2$ invariant of the decompletions as the solution to a finite system of recurrence equations.Comment: 18 pages, minor corrections according to referee comment

Degree distributions for a class of Circulant graphs

We characterize the equivalence and the weak equivalence of Cayley graphs for a finite group \C{A}. Using these characterizations, we find degree distribution polynomials for weak equivalence of some graphs including 1) circulant graphs of prime power order, 2) circulant graphs of order $4p$, 3) circulant graphs of square free order and 4) Cayley graphs of order $p$ or $2p$. As an application, we find an enumeration formula for the number of weak equivalence classes of circulant graphs of prime power order, order $4p$ and square free order and Cayley graphs of order $p$ or $2p$

Notes on Various Methods for Constructing Directed Strongly Regular Graphs

Duval, in "A Directed Graph Version of Strongly Regular Graphs" [{\it Journal of Combinatorial Theory}, Series A 47 (1988) 71 - 100], introduced the concept of directed strongly regular graphs. In this paper we construct several rich families of directed strongly regular graphs with new parameters. Our constructions yielding new parameters are based on extending known explicit constructions to cover more parameter sets. We also explore some of the links between Cayley graphs, block matrices and directed strongly regular graphs with certain parameters. Directed strongly regular graphs which are also Cayley graphs are interesting due to their having more algebraic structure. We construct directed strongly regular Cayley graphs with parameters $((m+1)s,ls,ld,ld-d,ld)$ where $d,l$ and $s$ are integers with $dm=ls$ and $1\leq l<m$. We also give a new block matrix characterization for directed strongly regular graphs with parameters $(m(dm+1),dm,m,m-1,m)$, which were first dicussed by Duval et al. in "Semidirect Product Constructions of Directed Strongly Regular Graphs" [{\it Journal of Combinatorial Theory}, Series A 104 (2003) 157 - 167]

Which weighted circulant networks have perfect state transfer?

The question of perfect state transfer existence in quantum spin networks based on weighted graphs has been recently presented by many authors. We give a simple condition for characterizing weighted circulant graphs allowing perfect state transfer in terms of their eigenvalues. This is done by extending the results about quantum periodicity existence in the networks obtained by Saxena, Severini and Shparlinski and characterizing integral graphs among weighted circulant graphs. Finally, classes of weighted circulant graphs supporting perfect state transfer are found. These classes completely cover the class of circulant graphs having perfect state transfer in the unweighted case. In fact, we show that there exists an weighted integral circulant graph with $n$ vertices having perfect state transfer if and only if $n$ is even. Moreover we prove the non-existence of perfect state transfer for several other classes of weighted integral circulant graphs of even order

Distance spectra and Distance energy of Integral Circulant Graphs

The distance energy of a graph $G$ is a recently developed energy-type invariant, defined as the sum of absolute values of the eigenvalues of the distance matrix of $G$. There was a vast research for the pairs and families of non-cospectral graphs having equal distance energy, and most of these constructions were based on the join of graphs. A graph is called circulant if it is Cayley graph on the circulant group, i.e. its adjacency matrix is circulant. A graph is called integral if all eigenvalues of its adjacency matrix are integers. Integral circulant graphs play an important role in modeling quantum spin networks supporting the perfect state transfer. In this paper, we characterize the distance spectra of integral circulant graphs and prove that these graphs have integral eigenvalues of distance matrix $D$. Furthermore, we calculate the distance spectra and distance energy of unitary Cayley graphs. In conclusion, we present two families of pairs $(G_1, G_2)$ of integral circulant graphs with equal distance energy -- in the first family $G_1$ is subgraph of $G_2$, while in the second family the diameter of both graphs is three.Comment: 11 pages, 1 figur

Quantum State Transfer on a Class of Circulant Graphs

We study the existence of quantum state transfer on non-integral circulant graphs. We find that continuous time quantum walks on quantum networks based on certain circulant graphs with $2^k$ $\left(k\in\mathbb{Z}\right)$ vertices exhibit pretty good state transfer when there is no external dynamic control over the system. We generalize few previously known results on pretty good state transfer on circulant graphs, and this way we re-discover all integral circulant graphs on $2^k$ vertices exhibiting perfect state transfer.Comment: arXiv admin note: text overlap with arXiv:1705.0885

On Jacobian group and complexity of I-graph I(n,k,l) through Chebyshev polynomials

We consider a family of I-graphs I(n,k,l), which is a generalization of the class of generalized Petersen graphs. In the present paper, we provide a new method for counting Jacobian group of the I-graph I(n,k,l). We show that the minimum number of generators of Jac(I(n,k,l)) is at least two and at most 2k + 2l - 1. Also, we obtain a closed formula for the number of spanning trees of I(n,k,l) in terms of Chebyshev polynomials. We investigate some arithmetical properties of this number and its asymptotic behaviour.Comment: 17. arXiv admin note: substantial text overlap with arXiv:1612.0337

When is a Tensor Product of Circulant Graphs Circulant?

In this paper we determine partial answers to the question given in the title, thereby significantly extending results of Broere and Hattingh. We characterize completely those pairs of complete graphs whose tensor products are circulant. We establish that if the orders of these circulant graphs have greatest common divisor of 2, the product is circulant whenever both graphs are bipartite. We also establish that it is possible for one of the two graphs not to be circulant and the product still to be circulant.Comment: 10 pages, LaTeX, no figure