Hamiltonicity of Cubic 3-Connected k-Halin Graphs

Abstract

We investigate here how far we can extend the notion of a Halin graph such that hamiltonicity is preserved. Let $H = T \cup C$ be a Halin graph, $T$ being a tree and $C$ the outer cycle. A $k$-Halin graph $G$ can be obtained from $H$ by adding edges while keeping planarity, joining vertices of $H - C$, such that $G - C$ has at most $k$ cycles. We prove that, in the class of cubic $3$-connected graphs, all $14$-Halin graphs are hamiltonian and all $7$-Halin graphs are $1$-edge hamiltonian. These results are best possible.

Commutativity and Compactness of Kth Order Slant Toeplitz Operators

“…The vastness of the field of Graph Theory can be perfectly utilized in highlighting some similarities and differences between the compactness and non compactness of k th order slant Toeplitz operators on H 2 . With that being said, several other results and theories can also be discussed by using various tools of Graph Theory [9], [14].…”

“…With the gradual advancement of graph theory in various spectra of mathematical and physical sciences, several results and theories about Toeplitz operators have also been analyzed and studied through graphical approach [9]. In [12], a new approach has been implemented to study some basic properties concerning the above operators, and with the notion of k th order slant Toeplitz graphs being introduced, a wide range of ideas has also been split open for the young enthusiasts in the field of Operator theory.…”