‎Let $G$ be a graph of order $n$ with vertices labeled as $v_1‎, ‎v_2,dots‎ , ‎v_n$‎. ‎Let $d_i$ be the degree of the vertex $v_i$ for $i = 1‎, ‎2‎, ‎cdots‎ , ‎n$‎. ‎The Albertson matrix of $G$ is the square matrix of order $n$ whose $(i‎, ‎j)$-entry is equal to $|d_i‎ - ‎d_j|$ if $v_i $ is adjacent to $v_j$ and zero‎, ‎otherwise‎. ‎The main purposes of this paper is to introduce the Albertson ...

Albertson conjectured that if a graph G has chromatic number r, then the crossing number of G is at least as large as the crossing number of Kr, the complete graph on r vertices. Albertson, Cranston, and Fox verified the conjecture for r 6 12. In this paper we prove it for r 6 16. Dedicated to the memory of Michael O. Albertson.

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 ...

<abstract><p>We introduce the general Albertson irregularity index of a connected graph $ G and define it as A_{p}(G) = (\sum_{uv\in E(G)}|d(u)-d(v)|^p)^{\frac{1}{p}} $, where p is positive real number d(v) degree vertex v in $. The new not only generalization well-known \sigma $-index, but also Minkowski norm vertex. We present lower upper bounds on index. In addition, we study ext...

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 ...

A distinguishing r-labeling of a digraph G is a mapping λ from the set of vertices of G to the set of labels {1, . . . , r} such that no nontrivial automorphism of G preserves all the labels. The distinguishing number D(G) of G is then the smallest r for which G admits a distinguishing r-labeling. From a result of Gluck (David Gluck, Trivial set-stabilizers in finite permutation groups, Can. J....

We prove that every triangle-free planar graph on n vertices with maximum degree three has an independent set with size at least 3 8 n. This was suggested and later conjectured by Albertson, Bollobás, and Tucker.

We prove that every triangle-free planar graph on n vertices with maximum degree three has an independent set with size at least 38n. This was suggested and later conjectured by Albertson, Bollobás, and Tucker.