Edge shift distance between isomorphism classes of graphs, introduced by M. Johnson, is investigated in the case of trees and compared with other distances.

There exists a bijection between one-stack sortable permutations (permutations which avoid the pattern (231)) and rooted plane trees. We define an edit distance between permutations which is consistent with the standard edit distance between trees. This one-to-one correspondence yields a polynomial algorithm for the subpermutation problem for (231) pattern-avoiding permutations. Moreover, we obtain the generating function of the edit distance between ordered unlabeled trees and some special ones. For...

The distance spectral radius ρ(G) of a graph G is the largest eigenvalue of the distance matrix D(G). Let U (n,m) be the class of unicyclic graphs of order n with given matching number m (m ≠ 3). In this paper, we determine the extremal unicyclic graph which has minimal distance spectral radius in U (n,m) Cn.

For two vertices $u$ and $v$ of a graph $G$, the closed interval $I[u,v]$ consists of $u$, $v$, and all vertices lying in some $u\text{--}v$ geodesic of $G$, while for $S\subseteq V\left(G\right)$, the set $I\left[S\right]$ is the union of all sets $I[u,v]$ for $u,v\in S$. A set $S$ of vertices of $G$ for which $I\left[S\right]=V\left(G\right)$ is a geodetic set for $G$, and the minimum cardinality of a geodetic set is the geodetic number $g\left(G\right)$. A vertex $v$ in $G$ is an extreme vertex if the subgraph induced by its neighborhood is complete. The number of extreme vertices in $G$ is its extreme order $\mathrm{e}x\left(G\right)$. A graph $G$ is an extreme geodesic...