Isomorphism Checking of I-graphs

Horvat, Boris; Pisanski, Tomaž; Žitnik, Arjana

doi:10.1007/s00373-011-1086-2

Cited by 15 publications

(18 citation statements)

References 12 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Therefore checking isomorphisms of I-graphs is important. The following result, recently proven in [17], see also [4], determines the collection of isomorphs of a given I-graph.…”

Section: Existence Of Admissible Isomorphs Of I-graphsmentioning

confidence: 92%

All Generalized Petersen Graphs Are Unit-Distance Graphs

Žitnik¹,

Horvat²,

Pisanski³

2012

Journal of the Korean Mathematical Society

Self Cite

View full text Add to dashboard Cite

Abstract. In 1950 a class of generalized Petersen graphs was introduced by Coxeter and around 1970 popularized by Frucht, Graver and Watkins. The family of I-graphs mentioned in 1988 by Bouwer et al. represents a slight further albeit important generalization of the renowned Petersen graph. We show that each I-graph I(n, j, k) admits a unit-distance representation in the Euclidean plane. This implies that each generalized Petersen graph admits a unit-distance representation in the Euclidean plane. In particular, we show that every I-graph I(n, j, k) has an isomorphic I-graph that admits a unit-distance representation in the Euclidean plane with a n-fold rotational symmetry, with the exception of the families I(n, j, j) and I(12m, m, 5m), m ≥ 1. We also provide unit-distance representations for these graphs.

show abstract

“…Therefore checking isomorphisms of I-graphs is important. The following result, recently proven in [17], see also [4], determines the collection of isomorphs of a given I-graph.…”

Section: Existence Of Admissible Isomorphs Of I-graphsmentioning

confidence: 92%

All Generalized Petersen Graphs Are Unit-Distance Graphs

Žitnik¹,

Horvat²,

Pisanski³

2012

Journal of the Korean Mathematical Society

Self Cite

View full text Add to dashboard Cite

show abstract

“…In [1], several graph-theoretic properties of I(n, j, k) such as connectedness, girth, being bipartite or being vertex-symmetric, are characterized in terms of number-theoretic properties of parameters n, j, k. An algorithm for deciding which sets of parameter values give rise to isomorphic I-graphs is also given there. In [5], the following result (crucial for our enumeration) is proved: Theorem 1.1. I(n, j, k) and I(n, j , k ) are isomorphic if and only if there exists an integer a, relatively prime to n, such that either {j , k } = {aj mod n, ak mod n} or {j , k } = {aj mod n, −ak mod n}.…”

Section: Introductionmentioning

confidence: 99%

Enumeration of I-graphs: Burnside does it again

Petkovšek

Zakrajšek

2009

Ars Math. Contemp.

View full text Add to dashboard Cite

show abstract

“…If GCD(n, k, l) = m > 1, then I(n, k, l) is a union of m copies of the graph I(n/m, k/m, l/m). If m = 1 and GCD(k, l) = d, then the graphs I(n, k, l) and I(n, k/d, l/d) are isomorphic [25], [22], [24]. So, in what follows, we assume k and l to be relatively prime.…”

Section: Introductionmentioning

confidence: 99%

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

Mednykh

2018

Ars Math. Contemp.

View full text Add to dashboard Cite

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.

show abstract

Isomorphism Checking of I-graphs

Cited by 15 publications

References 12 publications

All Generalized Petersen Graphs Are Unit-Distance Graphs

All Generalized Petersen Graphs Are Unit-Distance Graphs

Enumeration of I-graphs: Burnside does it again

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

Contact Info

Product

Resources

About