Seminar za biomatematiko in matematično kemijo - Arhiv

The transmission of a vertex in a connected graph is the sum of its distances to all the other vertices. A graph is transmission irregular (TI) when all of its vertices have mutually distinct transmissions. In an earlier paper, Al-Yakoob and Stevanović [Appl. Math. Comput. 380 (2020), 125257] gave the full characterization of TI starlike trees with three branches. Here, we improve these results by using a different approach to provide the complete characterization of all TI starlike trees and all TI double starlike trees. We subsequently implement the aforementioned conditions in order to find several infinite families of TI starlike trees and TI double starlike trees. Besides that, we disclose five families of unicyclic graphs with two pendent paths whose members are TI under certain conditions. As a direct consequence, we demonstrate the existence of TI chemical graphs of almost all even orders, thereby resolving a problem recently posed by Xu, Tian and Klavžar [Discrete Appl. Math. 340 (2023), 286–295].

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We characterize plane bipartite graphs whose resonance graphs are daisy cubes, and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we show that if G is a plane elementary bipartite graph other than K₂, then the resonance graph of G is a daisy cube if and only if the Fries number of G equals the number of finite faces of G. Next, we extend the above characterization from plane elementary bipartite graphs to plane bipartite graphs and show that the resonance graph of a plane bipartite graph G is a daisy cube if and only if G is weakly elementary bipartite such that each of its elementary component G_i other than K₂ holds the property that the Fries number of G_i equals the number of finite faces of G_i. Along the way, we provide a structural characterization for a plane elementary bipartite graph whose resonance graph is a daisy cube, and show that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes.

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