The diameter of a graph is the maximum over vertices u and v of the distance between u and v. It is easy to show that graphs with high conductance must have low diameter. If the set of edges leaving a set must be large relative to the size of the set, then after taking neighborhoods a few times, one must encounter most of the edges. We now make this intuition precise. Recall from Lecture 7 the ...

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. It is well known that for connected or disconnected graphs with n vertices and m edges, the inequality M2/m ≥ M1/n does not always hold. Here we show that this relati...

For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M2 is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we study the Zagreb indices of n-vertex connected graphs with k cut vertices, the upper bound for M1and M2-values of n-vertex connected graphs with k cut vertices are deter...

For a graph, the first Zagreb index M1 is equal to the sum of the squares of the degrees of the vertices, and the second Zagreb index M2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. Denote by Gn,k the set of graphs with n vertices and k cut edges. In this paper, we showed the types of graphs with the largest and the second largest M1 and M2 among Gn,k .

Throughout this paper, we assume 1 is a connected finite undirected graph without loops or multiple edges. We identify 1 with the set of vertices. For vertices : and ; in 1, let (:, ;) denote the distance between : and ; in 1, that is, the length of a shortest path connecting : and ;. Let d=d(1 ) denote the diameter of 1, that is, the maximal distance between any two vertices in 1. Let 1i (:)=[...

For a given graph G, the square of G, denoted by G2, is a graph with the vertex set V(G) such that two vertices are adjacent if and only if the distance of these vertices in G is at most two. A graph G is called squared if there exists some graph H such that G= H2. A function f:V(G) {0,1,2…, k} is called a coloring of G if for every pair of vertices x,yV(G) with d(x,y)=1 we have |f(x)-f(y)|2 an...

Solution. In a finite connected graph with n ≥ 2 vertices, the domain for the vertex degrees is the set {1, 2, . . . , n − 1} since each vertex can be adjacent to at most all of the remaining n−1 vertices and the existence of a degree 0 vertex would violate the assumption that the graph be connected. Therefore, treating the n vertices as the pigeons and the n−1 possible degrees as the pigeonhol...

The appropriate mathematical model for the problem space of tower transformation tasks is the state graph representing positions of discs or balls and their moves. Graph theoretical quantities like distance, eccentricities or degrees of vertices and symmetries of graphs support the choice of problems, the selection of tasks and the analysis of performance of subjects whose solution paths can be...

Universal scaling of distances between vertices of Erdos-Rényi random graphs, scale-free Barabási-Albert models, science collaboration networks, biological networks, Internet Autonomous Systems and public transport networks are observed. A mean distance between two nodes of degrees k(i) and k(j) equals to (l(ij)) = A - B log(k(i)k(j)). The scaling is valid over several decades. A simple theory ...