Nonseparating trees in 2-connected graphs and oriented trees in strongly connected digraphs

On spectral radius of strongly connected digraphs

 It is known that the directed cycle of order $n$ uniquely achieves the minimum spectral radius among all strongly connected digraphs of order $nge 3$. In this paper, among others, we determine the digraphs which achieve the second, the third and the fourth minimum spectral radii respectively among strongly connected digraphs of order $nge 4$.  

Nonseparating Planar Chains in 4-Connected Graphs

In this paper, we describe an O(|V (G)|) algorithm for finding a “non-separating planar chain” in a 4-connected graph G, which will be used to decompose an arbitrary 4-connected graph into “planar chains”. This work was motivated by the study of a multi-tree approach to reliability in distributed networks, as well as the study of non-separating induced paths in highly connected graphs. Supporte...

Strongly Connected Multivariate Digraphs

Generalizing the idea of viewing a digraph as a model of a linear map, we suggest a multi-variable analogue of a digraph, called a hydra, as a model of a multi-linear map. Walks in digraphs correspond to usual matrix multiplication while walks in hydras correspond to the tensor multiplication introduced by Robert Grone in 1987. By viewing matrix multiplication as a special case of this tensor m...

Non-separating trees in connected graphs

Let T be any tree of order d ≥ 1. We prove that every connected graph G with minimum degree d contains a subtree T ′ isomorphic to T such that G − V (T ) is connected.

Independent Trees in 4-Connected Graphs