On parametric equivalence, isomorphism and uniqueness: Cycle related graphs
ODAM-Vol. 4 (2021), Issue 1, pp. 45 – 51 Open Access Full-Text PDF
J. Kok, J. Shiny
Abstract: This furthers the notions of parametric equivalence, isomorphism and uniqueness in graphs. Results for certain cycle related graphs are presented. Avenues for further research are also suggested.
Kolakoski sequence: links between recurrence, symmetry and limit density
ODAM-Vol. 4 (2021), Issue 1, pp. 29 – 44 Open Access Full-Text PDF
Alessandro Della Corte
Abstract: The Kolakoski sequence \(S\) is the unique element of \(\left\lbrace 1,2 \right\rbrace^{\omega}\) starting with 1 and coinciding with its own run length encoding. We use the parity of the lengths of particular subclasses of initial words of \(S\) as a unifying tool to address the links between the main open questions – recurrence, mirror/reversal invariance and asymptotic density of digits. In particular we prove that recurrence implies reversal invariance, and give sufficient conditions which would imply that the density of 1s is \(\frac{1}{2}\).
Graph energy and nullity
ODAM-Vol. 4 (2021), Issue 1, pp. 25 – 28 Open Access Full-Text PDF
Ivan Gutman
Abstract: The energy of a graph is the sum of absolute values of its eigenvalues. The nullity of a graph is the algebraic multiplicity of number zero in its spectrum. Empirical facts indicate that graph energy decreases with increasing nullity, but proving this property is difficult. In this paper, a method is elaborated by means of which the effect of nullity on graph energy can be quantitatively estimated.
On parametric equivalent, isomorphic and unique sets
ODAM-Vol. 4 (2021), Issue 1, pp. 19 – 24 Open Access Full-Text PDF
J. Kok, J. Shiny
Abstract: This short paper introduces the notions of parametric equivalence, isomorphism and uniqueness in graphs. Results for paths, cycles and certain categories (or types) of trees with regards to minimum confluence sets are presented.
Deutsch paths and their enumeration
ODAM-Vol. 4 (2021), Issue 1, pp. 12 – 18 Open Access Full-Text PDF
Helmut Prodinger
Abstract: A variation of Dyck paths allows for down-steps of arbitrary length, not just one. Credits for this invention are given to Emeric Deutsch. Surprisingly, the enumeration of them is somewhat akin to the analysis of Motzkin-paths; the last section contains a bijection.
Dominator Chromatic numbers of orientations of trees
ODAM-Vol. 4 (2021), Issue 1, pp. 4 – 11 Open Access Full-Text PDF
Michael Cary
Abstract: In this paper we prove that the dominator chromatic number of every oriented tree is invariant under reversal of orientation. In addition to this marquee result, we also prove the exact dominator chromatic number for arborescences and anti-arborescences as well as bounds on other orientations of oft studied tree topologies including generalized stars and caterpillars.
Some basic properties of Sombor indices
ODAM-Vol. 4 (2021), Issue 1, pp. 1 – 3 Open Access Full-Text PDF
Ivan Gutman
Abstract: The recently introduced class of vertex-degree-based molecular structure descriptors, called Sombor indices (\(SO\)), are examined and a few of their basic properties established. Simple lower and upper bounds for \(SO\) are determined. It is shown that any vertex–degree–based descriptor can be viewed as a special case of a Sombor-type index.