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More on linear and metric tree maps

100%

Kozerenko S.

Opuscula Mathematica

2021

tom Vol. 41, no. 1

55--70

We consider linear and metric self-maps on vertex sets of finite combinatorial trees. Linear maps are maps which preserve intervals between pairs of vertices whereas metric maps are maps which do not increase distances between pairs of vertices. We obtain criteria for a given linear or a metric map to be a positive (negative) under some orientation of the edges in a tree, we characterize trees which admit maps with Markov graphs being paths and prove that the converse of any partial functional digraph is isomorphic to a Markov graph for some suitable map on a tree.

New results on imbalance graphic graphs

63%

Kozerenko S. , Serdiuk A.

tom Vol. 43, no. 1

81--100

An edge imbalance provides a local measure of how irregular a given graph is. In this paper, we study graphs with graphic imbalance sequences. We give a new proof of imbalance graphicness for trees and use the new idea to prove that the same holds for unicyclic graphs. We then show that antiregular graphs are imbalance graphic and consider the join operation on graphs as well as the double graph operation. Our main results are concerning imbalance graphicness of three classes of block graphs: block graphs having all cut vertices in a single block; block graphs in which the subgraph induced by the cut vertices is either a star or a path. In the end, we discuss open questions and conjectures regarding imbalance graphic graphs.

Graphs with odd and even distances between non-cut vertices

63%

Antoshyna K. , Kozerenko S.

Opuscula Mathematica

2025

tom Vol. 45, no. 1

5--25

We prove that in a connected graph, the distances between non-cut vertices are odd if and only if it is the line graph of a strong unique independence tree. We then show that any such tree can be inductively constructed from stars using a simple operation. Further, we study the connected graphs in which the distances between non-cut vertices are even (shortly, NCE-graphs). Our main results on NCE-graphs are the following: we give a criterion of NCE-graphs, show that any bipartite graph is an induced subgraph of an NCE-graph, characterize NCE-graphs with exactly two leaves, characterize graphs that can be subdivided to NCE-graphs, and provide a characterization for NCE-graphs which are maximal with respect to the edge addition operation.