Motivated by the problem of zero-error broadcasting, we introduce a new notion of graph capacity, termed ρ-capacity, that generalizes the Shannon capacity of a graph. We derive upper and lower bounds on the ρ-capacity of arbitrary graphs, and provide a Lovász-type upper bound for regular graphs. We study the behavior of the ρ-capacity under two graph operations: the strong product and the disjo...

Let $G$ be a finite group. Define graph on the set $G^{\#} = G \setminus \{ 1 \}$ by declaring distinct elements $x,y\in G^{\#}$ to adjacent if and only $\langle x,y\rangle$ is cyclic. Denote this $\Delta(G)$. The $\Delta(G)$ has appeared in literature under names cyclic deleted enhanced power graph. If $H$ are nontrivial groups, then $\Delta(G\times H)$ completely characterized. In particular,...

The standard products—cartesian, lexicographic, tensor, and strong—all belong to a class of products introduced by W. Imrich and H. Izbicki (1975, Monatsh. Math. 80, 277–281) and later called B-products by I. Broere and J. H. Hattingh (1990, Quaest. Math. 13, 191–216) who establish that the lexicographic product of two circulant graphs is again circulant. In this paper, we establish that any B-...

The center and the periphery of a graph is the set of vertices with minimum resp. maximum eccentricity in it. A graph is almost self-centered (ASC) if it contains exactly two non-central vertices and is almost peripheral (AP) if all its vertices but one lie in the periphery. Answering a question from (Taiwanese J. Math. 18 (2014), 463–471) it is proved that for any integer r ≥ 1 there exists an...