the first zagreb index $m_1$ of a graph $g$ is equal to the sum of squaresof degrees of the vertices of $g$. goubko proved that for trees with $n_1$pendent vertices, $m_1 geq 9,n_1-16$. we show how this result can beextended to hold for any connected graph with cyclomatic number $gamma geq 0$.in addition, graphs with $n$ vertices, $n_1$ pendent vertices, cyclomaticnumber $gamma$, and minimal $m...

Abstract. The L(2, 1)-labeling (or distance two labeling) of a graph G is an integer labeling of G in which two vertices at distance one from each other must have labels differing by at least 2 and those at distance two must differ by at least 1. The L(2, 1)labeling numberλ G of G is the smallest number k such that G has an L(2, 1)-labeling max{ ( ): ( )} f v v V G k ∈ = with max f v : v V G k....

We present some necessary and su cient conditions (in terms of degrees of vertices) for locally nite tree T=(V; E) to be paradoxical, i.e. to have a partition V=V1∪V2 and one-to-one mappings fi :V → Vi; i=1; 2 such that the supremum of the distances between v and fi(v) over v ∈ V and i = 1; 2 is nite. c © 2000 Published by Elsevier Science B.V. All rights reserved.

Random graph theory is used to examine the “smallworld phenomenon” – any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees, the average distance is almost surely of order log n/ log d̃ where d̃ is the weighted average of the sum of squares of the expected degrees. Of particular interest ar...

Random graph theory is used to examine the "small-world phenomenon"; any two strangers are connected through a short chain of mutual acquaintances. We will show that for certain families of random graphs with given expected degrees the average distance is almost surely of order log nlog d, where d is the weighted average of the sum of squares of the expected degrees. Of particular interest are ...