In a study on the structure–dependency of the total π-electron energy from 1972, Trinajstić and one of the present authors have shown that it depends on the sums ∑ v∈V d(v) 2 and ∑ v∈V d(v) , where d(v) is the degree of a vertex v of the underling molecular graph G. The first sum was later named first Zagreb index and over the years became one of the most investigated graph–based molecular stru...

The multiplicative sum Zagreb index is defined for a simple graph G as the product of the terms dG(u)+dG(v) over all edges uv∈E(G) , where dG(u) denotes the degree of the vertex u of G . In this paper, we present some lower bounds for the multiplicative sum Zagreb index of several graph operations such as union, join, corona product, composition, direct product, Cartesian product and strong pro...

In this paper, we propose a new method to solve the minimization problem in simultaneous reconstruction and segmentation (SRS) model for X-ray computed tomography (CT). The SRS uses Bayes' rule maximum posteriori (MAP) estimate on hidden Markov measure field (HMMFM). original [Romanov M, Dahl AB, Dong Y, Hansen PC. Simultaneous tomographic with class priors. Inverse Problems Sci Eng. 2016;24(8)...

Constraint toolkits generally propose a sum constraint where a global objective variable should be equal to a sum of local objective variables, on which bound-consistency is achieved. To solve optimization problems this propagation is poor. Therefore, ad-hoc techniques are designed for pruning the global objective variable by taking account of the interactions between constraints defined on loc...

Different clustering algorithms achieve different results to certain data sets because most clustering algorithms are sensitive to the input parameters and the structure of data sets. Cluster validity, as the way of evaluating the result of the clustering algorithms, is one of the problems in cluster analysis. In this paper, we build up a framework for cluster validity process, meanwhile a sum-...

Let c be a proper edge colouring of a graph G = (V,E) with integers 1, 2, . . . , k. Then k ≥ ∆(G), while by Vizing’s theorem, no more than k = ∆(G)+ 1 is necessary for constructing such c. On the course of investigating irregularities in graphs, it has beenmoreover conjectured that only slightly larger k, i.e., k = ∆(G) + 2 enables enforcing additional strong feature of c, namely that it attri...

For a given graph G, ε(v) and deg(v) denote the eccentricity and the degree of the vertex v in G, respectively. The adjacent eccentric distance sum index of a graph G is defined as [Formula in text], where [Formula in text] is the sum of all distances from the vertex v. In this paper we derive some bounds for the adjacent eccentric distance sum index in terms of some graph parameters, such as i...

an injective map f : e(g) → {±1, ±2, · · · , ±q} is said to be an edge pair sum labeling of a graph g(p, q) if the induced vertex function f*: v (g) → z − {0} defined by f*(v) = (sigma e∈ev) f (e) is one-one, where ev denotes the set of edges in g that are incident with a vetex v and f*(v (g)) is either of the form {±k1, ±k2, · · · , ±kp/2} or {±k1, ±k2, · · · , ±k(p−1)/2} u {k(p+1)/2} accordin...

The general sum-connectivity index of a graph $G$ is defined as $chi_{alpha}(G)= sum_{uvin E(G)} (d_u + d_{v})^{alpha}$ where $d_{u}$ is degree of the vertex $uin V(G)$, $alpha$ is a real number different from $0$ and $uv$ is the edge connecting the vertices $u,v$. In this note, the problem of characterizing the graphs having extremum $chi_{alpha}$ values from a certain collection of polyomino ...