We consider variants of the triangle-avoidance game first defined by Harary and rediscovered by Hajnal a few years later. A graph game begins with two players and an empty graph on n vertices. The two players take turns choosing edges within Kn, building up a simple graph. The edges must be chosen according to a set of restrictions R. The winner is the last player to choose an edge that does no...

In a paper of Harary and Plantholt, they concluded by noting that they knew of no generalization of the leaf edge exchange (LEE) transition sequence result on spanning trees to other natural families of spanning subgraphs. Now, we give two approaches for such a generalization. We define two kinds of LEE-graphs over the set of all connected spanning k-edge subgraphs of a connected graph G, and s...

Consider the Parzen-Rosenblatt kernel estimate f. = (l/n) ~ Kh (x-Xi), where h > 0 is a constant, K is an absolutely integrable i=1 function with integral one, Kh(X)=(1/ha)K(x/h), and X1 ..... X. are iid random variables with common density f on R d. We show that for all e > 0, n8 2 2 sup P(lSlf.-fl-E~lf.-fll>e)<Ze 32~1KI. whenever liminfl~E~lf,-f]=oo. We also study what happens when h is allow...

Some errors in literatures are pointed out and corrected. A generalization of Ostrowski type inequalities for functions whose derivatives in absolute value are s-convex in the second sense is established. Special cases are discussed.

A graph G=(V; E) is said to be an integral sum graph (sum graph) if its vertices can be given a labeling with distinct integers (positive integers), so that uv ∈ E if and only if u+ v ∈ V . The integral sum number (sum number) of a given graph G, denoted by (G) ( (G)), was de:ned as the smallest number of isolated vertices which when added to G result in an integral sum graph (sum graph). In th...

A quantification of the Frank and McFee-Parungao orthogonal electrocardiogram of 175 normal children, ages 2-10 years is presented. There are significant differences in age groups 2-5 and 6-10. The QRS voltages of the younger group demonstrate 1) greater Z anterior, 2) greater initial X right, 3) greater terminal X right, 4) smaller Y inferior, and 5) the T vector is oriented more posteriorly. ...

We discuss free subsemigroups in solvable and elementary amenable groups improving earlier results of Milnor-Wolf, Rosenblatt, Alperin and Osin about uniform exponential growth for these groups.

A fuzzy graph is utilized to characterize the role played by an individual member in such a group that a class of group members having relationship with any given member has no sharply defined boundary. The concepts of weakening and strengthening points of an ordinary graph presented by Ross and Harary are generalized to those of a fuzzy graph.