Randić index, R, also known as the connectivity or branching index, is an important topological index in chemistry. In order to attack some conjectures concerning Randić index, Dvořák et al. [5] introduced a modification of this index, denoted by R′. In this paper we present some of the basic properties of R′. We determine graphs with minimal and maximal values of R′, as well as graphs with min...

We present a graph-based variational algorithm for multiclass classification of high-dimensional data, motivated by total variation techniques. The energy functional is based on a diffuse interface model with a periodic potential. We augment the model by introducing an alternative measure of smoothness that preserves symmetry among the class labels. Through this modification of the standard Lap...

We propose a hybrid genetic algorithm for partitioning a VLSI circuit graph into two disjoint graphs of minimum cut size. The algorithm includes a local optimization heuristic which is a modification of Fiduccia-Matheyses algorithm. Using well-known benchmarks (including ACM/SIGDA benchmarks), the combination of genetic algorithm and the local heuristic performed better than hMetis, a represent...

We introduce a modification of the Harary index where the contributions of vertex pairs are weighted by the sum of their degrees. After establishing basic mathematical properties of the new invariant, we proceed by finding the extremal graphs and investigating its behavior under several standard graph products.

We provide a minimal counterexample to the correctness of an algorithm proposed by R. Tarjan for decomposing a graph by maximal clique separators. We also suggest a modification to that algorithm which not only corrects it but also retains its O(nm) time complexity.

We introduce data structures called responsibility set and kernel. We present an algorithm FCRK, which is a modification of FC that maintains these structures and uses them for pruning of the search space. According to our experimental evaluation, FC-RK outperforms FC-CBJ on constraint networks encoding graph k-coloring instances and on non-dense random binary constraint networks.

let $g$ be a non-abelian finite group. in this paper, we prove that $gamma(g)$ is $k_4$-free if and only if $g cong a times p$, where $a$ is an abelian group, $p$ is a $2$-group and $g/z(g) cong mathbb{ z}_2 times mathbb{z}_2$. also, we show that $gamma(g)$ is $k_{1,3}$-free if and only if $g cong {mathbb{s}}_3,~d_8$ or $q_8$.

we define minimal cn-dominating graph $mathbf {mcn}(g)$, commonality minimal cn-dominating graph $mathbf {cmcn}(g)$ and vertex minimal cn-dominating graph $mathbf {m_{v}cn}(g)$, characterizations are given for graph $g$ for which the newly defined graphs are connected. further serval new results are developed relating to these graphs.

In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

We investigate the tree gonality of a genus-g metric graph, defined as minimum degree tropical morphism from any modification graph to tree. give combinatorial constructive proof that this number is at most ⌈g/2⌉+1, fact whose proofs so far required an algebro-geometric detour via special divisors on curves. For even genus, which realizes bound belongs family morphisms pure dimension 3g−3 and h...