Abstract In this paper, the exact formulae for the Harary indices of join, disjunction, symmetric difference, strong product of graphs are obtained. Also, the Schultz and modified Schultz indices of join and strong product of graphs are computed. We apply some of our results to compute the Harary, Schultz and modified Schultz indices of fan graph, wheel graph, open fence and closed fence graphs.

The concept of balanced decomposition number was introduced by Fujita and Nakamigawa in connection with a simultaneous transfer problem. A balanced colouring of a graph G is a pair (R,B) of disjoint subsets R,B ⊆ V (G) with |R| = |B|. A balanced decomposition D of a balanced colouring C = (R,B) of G is a partition of vertices V (G) = V1 ∪ V2 ∪ . . . ∪ Vr such that G[Vi] is connected and |Vi ∩R|...

a graph g = (v,e) with p vertices and q edges is said to be a total mean cordial graph if there exists a function f : v (g) → {0, 1, 2} such that f(xy) = [(f(x)+f(y))/2] where x, y ∈ v (g), xy ∈ e(g), and the total number of 0, 1 and 2 are balanced. that is |evf (i) − evf (j)| ≤ 1, i, j ∈ {0, 1, 2} where evf (x) denotes the total number of vertices and edges labeled with x (x = 0, 1, 2). in thi...

Let G be a non-trivial, loopless graph and for each non-trivial subgraph H of G, let g(H) = |E(H)| |V(H)|−ω(H) . The graph G is 1-balanced if γ(G), the maximum among g(H), taken over all nontrivial subgraphs H of G, is attained when H = G. This quantity γ(G) is called the fractional arboricity of the graph G. The value γ(G) appears in a paper by Picard and Queyranne and has been studied extensi...

Graph complexity measures like tree-width, clique-width, NLC-width and rank-width are important because they yield Fixed Parameter Tractable algorithms. Rank-width is based on ranks of adjacency matrices of graphs over GF(2). We propose here algebraic operations on graphs that characterize rank-width. For algorithmic purposes, it is important to represent graphs by balanced terms. We give a uni...

The Wiener index W(G) of a connected graph G is defined as the sum of the distances between all unordered pairs of vertices of G. The eccentricity of a vertex v in G is the distance to a vertex farthest from v. In this paper we obtain the Wiener index of a graph in terms of eccentricities. Further we extend these results to the self-centered graphs.

In a distance-regular graph, the partition with respect to distance from a vertex supports a unique eigenvector for each eigenvalue. There may be non-singleton vertex sets whose corresponding distance partition also supports eigenvectors. We consider the members of three families of distance regular graphs, the Johnson Graphs, Hamming Graphs and Complete Multipartite graphs. For each we determi...

the corona product $gcirc h$ of two graphs $g$ and $h$ isobtained by taking one copy of $g$ and $|v(g)|$ copies of $h$;and by joining each vertex of the $i$-th copy of $h$ to the$i$-th vertex of $g$, where $1 leq i leq |v(g)|$. in thispaper, exact formulas for the eccentric distance sum and the edgerevised szeged indices of the corona product of graphs arepresented. we also study the conditions...

We present new serial and parallel algorithms for multilevel graph partitioning. Our algorithm has coarsening, partitioning and uncoarsening phases like other multilevel partitioning methods. However, we choose fixed nodes which are at least a specified distance away from each other and coarsen them with their neighbor nodes in the coarsening phase using various heuristics. Using this algorithm...

The balanced decomposition number f(G) of a graph G was introduced by Fujita and Nakamigawa [Discr. Appl. Math., 156 (2008), pp. 3339-3344]. A balanced colouring of a graph G is a colouring of some of the vertices of G with two colours, such that there is the same number of vertices in each colour. Then, f(G) is the minimum integer s with the following property: For any balanced colouring of G,...