Next to the shortest path distance, the second most popular distance function between vertices in a graph is the commute distance (resistance distance). For two vertices u and v, the hitting time Huv is the expected time it takes a random walk to travel from u to v. The commute time is its symmetrized version Cuv = Huv + Hvu. In our paper we study the behavior of hitting times and commute dista...

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...

‎the gutman index and degree distance of a connected graph $g$ are defined as‎ ‎begin{eqnarray*}‎ ‎textrm{gut}(g)=sum_{{u,v}subseteq v(g)}d(u)d(v)d_g(u,v)‎, ‎end{eqnarray*}‎ ‎and‎ ‎begin{eqnarray*}‎ ‎dd(g)=sum_{{u,v}subseteq v(g)}(d(u)+d(v))d_g(u,v)‎, ‎end{eqnarray*}‎ ‎respectively‎, ‎where‎ ‎$d(u)$ is the degree of vertex $u$ and $d_g(u,v)$ is the distance between vertices $u$ and $v$‎. ‎in th...

In this paper, we propose a novel interactive image segmentation method for RGB-D images using hierarchical Graph Cut. Considering the characteristics of RGB channels and depth channel in RGB-D image, we utilize Euclidean distance on RGB space and geodesic distance on 3D space to measure how likely a pixel belongs to foreground or background in color and depth respectively, and integrate the co...

Let G be a connected graph of order n. The resistance matrix of G is defined as RG = (rij(G))n×n, where rij(G) is the resistance distance between two vertices i and j in G. Eigenvalues of RG are called R-eigenvalues of G. If all row sums of RG are equal, then G is called resistance-regular. For any connected graph G, we show that RG determines the structure of G up to isomorphism. Moreover, the...

Let G be a simple connected graph. The generalized polarity Wiener index of G is defined as the number of unordered pairs of vertices of G whose distance is k. Some formulas are obtained for computing the generalized polarity Wiener index of the Cartesian product and the tensor product of graphs in this article.

Professor Frank Harary has had a singular influence on graph theory through his own extensive original research, through the training of several researchers who themselves have made many important contributions, and through popularizing work, most significantly his text Graph Theory [1]. Frank Harary has contributed to a number of more advanced specialized texts, like that of Buckley and Harary...

‎Let G=(V(G),E(G)) be a simple connected graph with vertex set V(G) and edge‎ ‎set E(G)‎. ‎The (first) edge-hyper Wiener index of the graph G is defined as‎: ‎$$WW_{e}(G)=sum_{{f,g}subseteq E(G)}(d_{e}(f,g|G)+d_{e}^{2}(f,g|G))=frac{1}{2}sum_{fin E(G)}(d_{e}(f|G)+d^{2}_{e}(f|G)),$$‎ ‎where de(f,g|G) denotes the distance between the edges f=xy and g=uv in E(G) and de(f|G)=∑g€(G)de(f,g|G). ‎In thi...

it is necessary to generate the automorphism group of a chemical graph in computer-aidedstructure elucidation. an euclidean graph associated with a molecule is defined by a weightedgraph with adjacency matrix m = [dij], where for i≠j, dij is the euclidean distance between thenuclei i and j. in this matrix dii can be taken as zero if all the nuclei are equivalent. otherwise,one may introduce dif...