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 ...

Using graph-theoretical techniques, we establish an inequality regarding the number of walks and closed walks in a graph. This inequality yields several upper bounds for the number of closed walks in a graph in terms of the number of vertices, number of edges, maximum degree, degree sequence, and the Zagreb indices of the graph. As applications, we also present some new upper bounds on the Estr...

In this note we examine the number of integer lattice paths consisting of up-steps (1, 1) and down-steps (1,−1) that do not touch the lines y = m and y = −k, and in particular Theorem 3.2 in [P. Mladenović, Combinatorics, Mathematical Society of Serbia, Belgrade, 2001]. The theorem is shown to be incorrect for n > m + k + min(m, k), and using similar combinatorial technique we proved the upper ...

Abstract. In this document we summarize a few supplementary results to the accompanying paper. We give a detailed proof of the bounds on the normalized total communicability obtained via quadrature rules (Theorem 3.1, section 3). We also derive the computational costs for the heuristics introduced. Moreover, this document contains the results of some numerical experiments performed on four smal...

For a graph G, Albertson [1] has defined the irregularity of G as

For every d ≥ 3 and k ∈ {2} ∪ [3,∞), we determine the smallest ε such that every fractional (k + ε)-precoloring of vertices at mutual distance at least d of a graph G with fractional chromatic number equal to k can be extended to a proper fractional (k + ε)-coloring of G. Our work complements the analogous results of Albertson for ordinary colorings and those of Albertson and West for circular ...

Hester, Thomas R. 1973 Chronological ordering of Great Basin prehistory. University of California Archaeological Research Facility, Contributions 17:17-18. Johnson, L., Jr. 1969 Obsidian hydration rate for the Klamath Basin of California and Oregon. Science 1965:1354-56. Katsui, Yoshio, and Tuko Kondo 1964 Dating of stone implements by using hydration layer of obsidian. Department of Geology an...

Human cancers are heterogeneous due to combined effects of genetic instability and selection, where the accumulation of the most advantageous set of genetic aberrations results in the expansion of cancer cells (Pinkel & Albertson, 2005). There are many different types of instability that occurs during tumor development, such as point mutation, alteration of microsatellite sequences, chromosome ...