Journal of Prime Research in Mathematics

# Realizable degree sequences of inner dual graphs of benzenoid systems

Faqir M. Bhatti$$^a$$,Hasan Baloch$$^{a,*}$$,Mehar Ali Malik$$^b$$,Rameez Ragheb$$^c$$

$$^a$$Riphah Institute of Computing and Applied Sciences (RICAS), Riphah International University, QIE Lahore, Pakistan.

$$^a$$Center for Advanced Studies in Mathematics (CASM), Lahore University of Management Sciences (LUMS) DHA, Lahore, Pakistan.

$$^a$$Dhanani School of Science & Engineering, Habib University, Karachi, Pakistan.

Correspondence should be addressed to: fmbhatti@riphah.edu.pk

### Abstract

An inner dual graph of a planar rigid benzenoid (hexagonal) system is a subgraph of the triangular lattice with the constraint that any two adjacent faces in the corresponding hexagonal system must be connected via an edge in the inner dual. The maximum degree of any vertex in an inner dual graph of a hexagonal system is 6. In contrast with the already existing algorithms in the literature that are used to check a given degree sequence to be graphically realizable, in this paper, we use a a simple technique to check the realizable degree sequences of inner dual graphs of benzenoid systems that form a rich class of molecular graphs in theoretical chemistry. We restrict the maximum degree to 3 and identify, by providing necessary and sufficient conditions on the values of α, β and γ, all the degree sequences of the form d = (3α, 2β, 1γ) that are graphically (inner dual of planar rigid hexagonal system) realizable. That is, we provide general constructions of the graphs (inner dual of planar rigid hexagonal system) realizing the degree sequences of the form d = (3α, 2β, 1γ).

#### Keywords:

Hexagonal system, Inner dual graph, Matchstick graph, Degree sequence, Graphical realization, Graphical sequence.