Search

Let G be a molecular graph with n vertices, m edges and G(L) be a line graph. Both G and L can be represented by their adjacency matrices A and E, respectively. The eigenvalues of G and L are denoted by, λ 1 ⩾ λ 2 ⩾ ... ⩾ λ n and γ 1 ⩾ γ 2 ⩾ ... ⩾ γ n , respectively. The Estrada index A and L are defined as E E (G) = ∑ i = 1 n e λ i and E E (L) = ∑ i = 1 n e γ i . In this article, we present some new lower and upper bounds for the Estrada index molecular and line graphs of phenylenes. [ABSTRACT FROM AUTHOR]

A wide range of new drugs emerge each year with the rapid development of the manufacture of medicines. Once the indicators of these drug molecular structures are calculated in view of defining the topological indices, these can be used to understand their medical properties. In this paper, based on the drug molecular structure analysis, we compute the multiplicative sum connectivity index, the multiplicative Randić index, the multiplicative atomic bond connectivity index, the multiplicative Geometric arithmetic index, the multiplicative Harmonic index and the multiplicative augmented Zagreb index for the molecular structure anticancer drugs SP[n]. Finally, we used Maple 15 to plot surfaces associated to compare our results. [ABSTRACT FROM AUTHOR]

An acyclic colouring of a graph is a proper colouring of the graph, for which every cycle uses at least three colours. In this paper, we describe a method for constructing all 4-connected acyclically 4-colourable planar triangulations that have exactly four odd-vertices, except the ones that contain no adjacent odd-vertices. Unlike previous operations, our method successfully establishes a connection with (acyclic) 4-colourings. Moreover, we discuss a special class of graphs, called diamond triangulations, and give a necessary and sufficient condition for a diamond triangulation to be acyclically 4-colourable. [ABSTRACT FROM AUTHOR]

An outer-independent k-rainbow dominating function of a graph G is a function f from V (G) to the set of all subsets of { 1 , 2 , ... , k } such that both the following hold: (i) { 1 , ... , k } = ⋃ u ∈ N (v) f (u) whenever v is a vertex with f (v) = ∅ , and (ii) the set of all v ∈ V (G) with f (v) = ∅ is independent. The outer-independent k-rainbow domination number of G is the invariant γ o i r k (G) , which is the minimum sum (over all the vertices of G) of the cardinalities of the subsets assigned by an outer-independent k-rainbow dominating function. In this paper, we initiate the study of outer-independent k-rainbow domination. We first investigate the basic properties of the outer-independent k-rainbow domination and then we focus on the outer-independent 2-rainbow domination number and present sharp lower and upper bounds for it. [ABSTRACT FROM AUTHOR]

A graphGis uniquelyk-colourable if the chromatic number ofGiskandGhas only onek-colouring up to permutation of the colours. Aksionov [On uniquely 3-colorable planar graphs, Discrete Math. 20 (1977), pp. 209–216] conjectured that every uniquely 3-colourable planar graph with at least four vertices has two adjacent triangles. However, in the same year, Melnikov and Steinberg [L.S. Mel'nikov and R. Steinberg,One counterexample for two conjectures on three coloring, Discrete Math. 20 (1977), pp. 203–206.] disproved the conjecture by constructing a counterexample. In this paper, we prove that if a uniquely 3-colourable planar graphGhas at most 4 triangles thenGhas two adjacent triangles. Furthermore, for any, we construct a uniquely 3-colourable planar graph withktriangles and without adjacent triangles. [ABSTRACT FROM PUBLISHER]

A subsetDof the vertex set of a graphGis asecure dominating setofGifDis a dominating set ofGand if, for each vertexunot inD, there is a vertexvinDadjacent tousuch that the swap set (D \{v}) ∪ {u} is again a dominating set ofG. Thesecure domination numberofG, denoted byγs(G), is the cardinality of a smallest secure dominating set ofG. In this paper, we prove that for any treeTonn ≥3 vertices,and the bounds are sharp, whereℓandtare the numbers of leaves and stems ofT, respectively. Moreover, we characterize the treesTsuch that. [ABSTRACT FROM AUTHOR]

A local colouring of a graphGis a functionc: V(G)→ℕ such that for eachS⊆V(G), 2≤|S|≤3, there existu, v∈Swith |c(u)−c(v)| at least the number of edges in the subgraph induced byS. The maximum colour assigned bycis the value χℓ(c) ofc, and the local chromatic number ofGis χℓ(G)=min {χℓ(c):cis a local colouring ofG}. In this note the local chromatic number is determined for Cartesian productsG □ H, whereGandGHare 3-colourable graphs. This result in part corrects an error from Omoomi and Pourmiri [On the local colourings of graphs, Ars Combin. 86 (2008), pp. 147–159]. It is also proved that ifGandHare graphs such that χ(G)≤⌊ χℓ(H)/2 ⌋, then χℓ(G □ H)≤χℓ(H)+1. [ABSTRACT FROM AUTHOR]