A new family of distances for graph vertices is proposed. These distances reduce to the shortest path distance and to the resistance distance at the extreme values of the family parameter. The most important property of them is that they are graphgeodetic: d(i, j)+d(j, k) = d(i, k) if and only if every path from i to k passes through j. The construction of the distances is based on the matrix f...

let $a = (a_{i,j})_{1 leq i,j leq n}$ be an $n times n$ matrixwhere $n geq 2$. let $dt(a)$, its second immanant be the immanant corresponding to the partition $lambda_2 = 2,1^{n-2}$. let $g$ be a connected graph with blocks $b_1, b_2, ldots b_p$ and with$q$-exponential distance matrix $ed_g$. we given an explicitformula for $dt(ed_g)$ which shows that $dt(ed_g)$ is independent of the manner in ...

A set $Wsubset V (G)$ is called a resolving set, if for every two distinct vertices $u, v in V (G)$ there exists $win W$ such that $d(u,w) not = d(v,w)$, where $d(x, y)$ is the distance between the vertices $x$ and $y$. A resolving set for $G$ with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we establish...

We define and study two new kinds of “effective resistances” based on hubs-biased – hubs-repelling hubs-attracting models navigating a graph/network. prove that these effective resistances are squared Euclidean distances between the vertices graph. They can be expressed in terms Moore–Penrose pseudoinverse Laplacian matrices analogous Kirchhoff indices graph resistance distances. several result...

• We adapt a traditional non-learnable GED algorithm to the novel paradigm of geometric deep learning. Triplet network for learning graph distances by means neural networks. Learning distance in domain without an embedding stage. Graph-based keyword spotting application with state-of-the-art performance. The emergence as framework deal graph-based representations has faded away approaches favor...

let g = (v, e) be a simple graph. hosoya polynomial of g isd(u,v)h(g, x) = {u,v}v(g)x , where, d(u ,v) denotes the distance between vertices uand v. as is the case with other graph polynomials, such as chromatic, independence anddomination polynomial, it is natural to study the roots of hosoya polynomial of a graph. inthis paper we study the roots of hosoya polynomials of some specific graphs.

Relative centricity RC values of vertices/atoms are calculated within the Distance Detour and Cluj-Distance criteria on their corresponding Shell transforms. The vertex RC distribution in a molecular graph gives atom equivalence classes, useful in interpretation of NMR spectra. Timed by vertex valences, RC provides a new index, called Centric Connectivity CC, which can be useful in the topologi...

The Kirchhoff index of a connected graph is the sum of resistance distances between all unordered pairs of vertices in the graph. Its considerable applications are found in a variety of fields. In this paper, we determine the maximum value of Kirchhoff index among the unicyclic graphs with fixed number of vertices and maximum degree, and characterize the corresponding extremal graph.

We introduce the notion of uniform number of a graph. The  uniform number of a connected graph $G$ is the least cardinality of a nonempty subset $M$ of the vertex set of $G$ for which the function $f_M: M^crightarrow mathcal{P}(X) - {emptyset}$ defined as $f_M(x) = {D(x, y): y in M}$ is a constant function, where $D(x, y)$ is the detour distance between $x$ and $y$ in $G$ and $mathcal{P}(X)$ ...