Jozsef Szilagyi*, Marc B. Parlange, Gabriel G. Katul & John D. Albertson Conservation and Survey Division, Institute of Agriculture and Natural Resources, University of Nebraska-Lincoln, 113 Nebraska Hall, Lincoln, NE 68588-0517, USA Department of Geography and Environmental Engineering, The Johns Hopkins University, Baltimore, MD, 21218-2686, USA School of the Environment, Duke University, Dur...

A graph $G$ is called irregular if the degrees of all its vertices are not same. said to be \textit{Stepwise Irregular} (SI) difference any two adjacent always 1 (one). This paper deals with \textit{2-Stepwise (2-SI) graphs in which every pair differ by 2. Here we discuss some properties 2-SI and generalize them for $k$-SI imbalance edge $k$. Besides, also compute bounds irregularity Albertson ...

For .f a set of graphs, define the bounded chromatic number Xb(&) (resp. the bounded path chromatic number Xpfë) ) to be the minimum number of colors c for which there exists a constant M such that every graph G € 2? can be vertex c-colored so that all but M vertices of G are properly colored (resp. the length of the longest monochromatic path in G is at most M ). For 3? the set of toroidal gra...

Chemical graphs, as other ones, are regular if all their vertices have the same degree. Otherwise, they are irregular, and it is of interest to measure their irregularity both for descriptive purposes and for QSAR/QSPR studies. Three indices have been proposed in the literature for that purpose: those of Collatz-Sinogowitz, of Albertson, and of Bell's variance of degrees. We study their propert...

A graph G is said to be hom-idempotent if there is a homomorphism from G2 to G, and weakly hom-idempotent if for some n ≥ 1 there is a homomorphism from Gn+1 to Gn . We characterize both classes of graphs in terms of a special class of Cayley graphs called normal Cayley graphs. This allows us to construct, for any integer n, a Cayley graph G such that Gn+1 → Gn 6→ Gn−1, answering a question of ...

We consider only finite undirected graphs without loops or multiple edges. For terminology and notation not defined here we refer the reader to Bondy and Murty’s textbook [6]. In this note we investigate irregularity measures of graphs. A graph is regular if all its vertices have the same degree, otherwise it is irregular. However, it is of interest to measure how irregular it is. Several appro...

In a graph G of maximum degree ∆ let γ denote the largest fraction of edges that can be ∆ edge-coloured. Albertson and Haas showed that γ ≥ 13 15 when G is cubic . We show here that this result can be extended to graphs with maximum degree 3 with the exception of a graph on 5 vertices. Moreover, there are exactly two graphs with maximum degree 3 (one being obviously the Petersen graph) for whic...

A spanning tree T of a graph G is called a homeomorphically irreducible spanning tree (HIST) if T does not contain vertices of degree 2. A graph G is called locally connected if for every vertex v ∈ V (G), the subgraph induced by the neighborhood of v is connected. In this paper, we prove that every connected and locally connected graph with more than 3 vertices contains a HIST. Consequently, w...