A (1, 2)-eulerian weight w of a cubic graph is called a Hamilton weight if every faithful circuit cover of the graph with respect to w is a set of two Hamilton circuits. Let G be a 3-connected cubic graph containing no Petersen minor. It is proved in this paper that G admits a Hamilton weight if and only if G can be obtained from K4 by a series of4$Y-operations. As a byproduct of the proof of t...

In a public facility or private office where many people can get together, fire detection device is mandatory tool as an emergency alarm in the facility. However, expense of installation troublesome matter. So, optimization needed to minimize number these devices. The way implement select appropriate position place devices facilities. research discussed placement sensors multi-story buildings. ...

A product network defines a class of topologies that are very often used such as meshes, tori, and hypercubes, etc. This paper proposes a generalized algorithm for fault-tolerant parallel sorting in product networks. To tolerate r 1 faulty nodes, an r-dimensional product network containing faulty nodes is partitioned into a number of subgraphs such that each subgraph contains at most one fault....

A product network defines a class of topologies that are very often used such as meshes, tori, and hypercubes, etc. This paper proposes a generalized algorithm for fault-tolerant parallel sorting in product networks. To tolerate r−1 faulty nodes, an r-dimensional product network containing faulty nodes is partitioned into a number of subgraphs such that each subgraph contains at most one fault....

A family of connected graphs G is said to be a family with constant metric dimension if its metric dimension is finite and does not depend upon the choice of G in G. In this paper, we study the metric dimension of the generalized Petersen graphs P (n, m) for n = 2m + 1 and m ≥ 1 and give partial answer of the question raised in [9]: Is P (n, m) for n ≥ 7 and 3 ≤ m ≤ bn−1 2 c, a family of graphs...

In line with symmetrical graphs such as Cayley graphs and vertex transitive graphs, we introduce a new class of symmetrical graphs called diametrically uniform graphs. The class of diametrically uniform graphs includes vertex transitive graphs and hence Cayley graphs. A tree t-spanner of graph G is a spanning tree T in which the distance between every pair of vertices is at most t times their d...

A double Roman dominating function on a graph G=(V,E) is f:V?{0,1,2,3} satisfying the condition that every vertex u for which f(u)=0 adjacent to at least one assigned 3 or two vertices 2, and with f(u)=1 2 3. The weight of f equals w(f)=?v?Vf(v). domination number ?dR(G) G minimum G. We obtain closed expressions generalized Petersen graphs P(5k,k). It proven ?dR(P(5k,k))=8k k?2,3mod5 8k??dR(P(5...

A general method for finding elementary abelian regular covering projections of finite connected graphs is applied to the Petersen graph. As a result, a complete list of pairwise nonisomophic elementary abelian covers admitting a lift of a vertex-transitive group of automorphisms is given. The resulting graphs are explicitly described in terms of voltage assignments.