We examine the influence of heterogeneous curing rates for a SIS model, used for malware spreading on the Internet, information dissemination in unreliable networks, and propagation of failures in networks. The topology structures considered are the regular graph which represents the homogenous network structures and the complete bi-partite graph which represents the hierarchical network struct...

In many applications in sequencing and scheduling it is desirable to have an underlaying graph as equitably colored as possible. In this paper, we consider an equitable coloring of some corona products H G of two graphs G and H. In particular, we show that deciding the colorability of H G is NP-complete even if G is 4-regular and H is . 2 K Next, we prove exact values or upper bounds on the equ...

The list Ramsey number R ℓ ( H , k ) ${R}_{\ell }(H,k)$ recently introduced by Alon, Bucić, Kalvari, Kuperwasser, and Szabó, is a list-coloring variant of the classical number. They showed that if $H$ fixed r $r$ -uniform hypergraph not -partite colors $k$ goes to infinity, e Ω ≤ O ${e}^{{\rm{\Omega }}(\sqrt{k})}\le {R}_{\ell }(H,k)\le {e}^{O(k)}$ . We prove = Θ }(H,k)={e}^{{\rm{\Theta }}(k)}$ ...

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...

We introduce a special type of graph embedding called book embedding and apply it to posets. A book embedding scheme for bipartite graphs is given, and is used to extend the embeddings to general k-partite graphs. Finally, we view the Hasse diagram of a poset as a directed k-partite graph and use this scheme to derive a book embedding for arbitrary posets. Using this book embedding scheme, we a...

In a connected graph G, the status of a vertex is the sum of the distances of that vertex to each of the other vertices in G. The subgraph induced by the vertices of minimum (maximum) status in G is called the median (anti-median) of G. The median problem of graphs is closely related to the optimization problems involving the placement of network servers, the core of the entire networks. Bipart...

Let P (G, λ) be the chromatic polynomial of a graph G. A graph G is chromatically unique if for any graph H, P (H, λ) = P (G, λ) implies H is isomorphic to G. Liu et al. [Liu, R. Y., Zhao, H. X., Ye, C. F.: A complete solution to a conjecture on chromatic uniqueness of complete tripartite graphs. Discrete Math., 289, 175–179 (2004)], and Lau and Peng [Lau, G. C., Peng, Y. H.: Chromatic uniquene...