Let $G$ be a simple graph of order $n$ and size $m$.The edge covering of $G$ is a set of edges such that every vertex of $G$ is incident to at least one edge of the set. The edge cover polynomial of $G$ is the polynomial$E(G,x)=sum_{i=rho(G)}^{m} e(G,i) x^{i}$,where $e(G,i)$ is the number of edge coverings of $G$ of size $i$, and$rho(G)$ is the edge covering number of $G$. In this paper we stud...

The packing chromatic number χρ(G) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets Π1, . . . ,Πk, where Πi, i ∈ [k], is an i-packing. The following conjecture is posed and studied: if G is a subcubic graph, then χρ(S(G)) ≤ 5, where S(G) is the subdivision of G. The conjecture is proved for all generalized prisms of cycles. To get this result it ...

Using voltage graphs, we construct the the smallest known trivalent graph of girth 16. The graph has 960 vertices and is a lift of the Petersen graph. By way of preparation, other lifts of the Petersen graph are discussed. AMS Subject Classifications: 05C25, 05C35

The distance matrix of a simple graph G is D(G) = (di,j), where di,j is the distance between the ith and jth vertices of G. The greatest eigenvalue λ1 of D(G) is called the distance spectral radius of the graph G and is denoted by λ1(G). A simple connected graph G is called a 2-partite distance regular graph if there exists a partition V1 ∪V2 of the vertex set of G such that for i = 1, 2 and an...

The main contributions of this paper are three-fold. First, we use a dynamic approach based on Reiter’s pioneering work on Karp-Miller computation graphs [19] to give a new and short proof of Mohar’s Minty-type Theorem [15]. Second, we bridge circular colorings and discrete event dynamic systems to show that the Barbosa and Gafni’s results on circular chromatic number [5, 21] can be generalized...

Paul Erdos defined the concept of coprime graph and studied about cycles in coprime graphs. In this paper this concept is generalized and a new graph called Generalized coprime graph is introduced. Having observed certain basic properties of the new graph it is proved that the chromatic number and the clique number of some generalized coprime graphs are equal.

The Petersen graph is a Moore graph that has node degree 3, diameter 2, and optimal network size 10. In this paper, we present a class of interconnection networks, called cyclic Petersen networks (CPNs), which efficiently extend the Petersen graph to obtain larger networks with small diameter and node degree. We derive balanced routing algorithms and efficient embeddings for CPNs. In particular...