Journal of Interconnection Networks

Let R be a finite commutative ring with identity. The co-maximal graph Γ(R) is a graph with vertices as elements of R, where two distinct vertices a and b are adjacent if and only if Ra + Rb = R. Also, Γ₂(R) is the subgraph of Γ(R) induced by non-unit elements and ${{\Gamma }^{\prime }}_2\left(R\right)={\Gamma }_2\left(R\right)\backslash J\left(R\right)$ where J(R) is Jacobson radical. In this paper, we characterize the rings for which the graphs Γ(R) and L(Γ(R)) are planar. Also, we characterize rings for which ${{\Gamma }^{\prime }}_2\left(R\right)$, Γ(R) and L(Γ(R)) are outerplanar along with some domination parameters on co-maximal graph.

Consider a network D of pipes which have to be cleaned using some cleaning agents, called brushes, assigned to some vertices. The minimum number of brushes required for cleaning the network D is called its brush number. The tattooing of a simple connected directed graph D is a particular type of the cleaning in which an arc are coloured by the colour of the colour-brush transiting it and the tattoo number of D is a corresponding derivative of brush numbers in it. Tattooing along an out-arc of a vertex v may proceed if a minimum set of colour-brushes is allocated (primary colours) or combined with those which have arrived (including colour blends) together with mutation of permissible new colour blends, has cardinality greater than or equal to $d_G^{+}\left(v\right)$.

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color classes. The b-chromatic number of a graph G, denoted by φ(G), is the largest integer k such that G has a b-coloring with k colors. The b-chromatic sum of a graph G(V, E), denoted by φ′(G) is defined as the minimum of sum of colors c(v) of v for all v ∈ V in a b-coloring of G using φ(G) colors. The Mycielskian or Mycielski, μ(H) of a graph H with vertex set {v₁, v₂,…, v_n} is a graph G obtained from H by adding a set of n + 1 new vertices {u, u₁, u₂, …, u_n} joining u to each vertex u_i(1 ≤ i ≤ n) and joining u_i to each neighbour of v_i in H. In this paper, the b-chromatic sum of Mycielskian of cycles, complete graphs and complete bipartite graphs are discussed. Also, an application of b-coloring in image processing is discussed here.

While designing router micro-architecture of an On-Chip network, a good allocation of virtual channels (VCs) governs an effective resources utilization which essentially results in an optimized number of packets received at destination(s). Generally, the VC allocation schemes deal with the one-way approach of VC allocation to the contending flits. However, this approach produces non-optimal matching of flits to the available VCs on next routers, and therefore leads to the under-utilization of these VCs. This paper proposes a 2-Way VC Allocation scheme to map input VCs (requestors) to output VCs (resources). The proposed scheme is compared with the conventional VC allocation scheme under two different mesh configurations with a 100% channel load. Simulations performed under two different routing schemes in diverse traffic scenarios demonstrate an increase in the number of packets received at destinations by up to 76%. Also, the network’s latency exhibits trade-off with total power consumption while reducing hotspots.