Du, Zhou and Trinajstić [J. Math. Chem., 48, pp. 697–703, 2010] proved that in the set of n-vertex connected unicyclic graphs G with n ≥ 3, for –1 ≤ α < 0, the graph consisting of a triangle and n – 3 pendant vertices adjacent to the same vertex of this triangle is the unique graph with the minimum general sum-connectivity index. In this paper, using a graph transformation and several inequalit...

The Randić index of a graph G is defined as R(G) = ∑ u∼v (d(u)d(v))− 2 , where d(u) is the degree of vertex u, and the summation goes over all pairs of adjacent vertices u, v. A conjecture of R(G) for connected graph with n vertices is as follows: R(G)−D √2− n+1 2 and R(G) D 1 2 + √ 2−1 n−1 , where D is the diameter of G. In this paper, we prove that this conjecture is true for unicyclic graphs.

Let R be a commutative ring with non-zero identity. The cozero-divisor graph of R, denoted by Γ′(R), is a graph with vertices in W ∗(R), which is the set of all non-zero and non-unit elements of R, and two distinct vertices a and b in W ∗(R) are adjacent if and only if a / ∈ bR and b / ∈ aR. In this paper, we characterize all commutative rings whose cozero-divisor graphs are forest, star, doubl...

Let G(V, E) be a graph. A vertex v ∈ V(G) is said to be a self vertex switching of G, if G is isomorphic to G, where G is the graph obtained from G, by deleting all edges of G incident to v and adding edges between v and the vertices which are not adjacent to v in G. In this paper, we discuss some applications of self vertex switching and list out all trees and unicyclic graphs with unique self...

The harmonic index of a graph G is defined as the sum of the weights 2 d.u/Cd.v/ of all edges uv of G, where d.u/ denotes the degree of a vertex u in G. In this paper, we present the minimum harmonic indices for unicyclic and bicyclic graphs with n vertices and matching number m (2 m bn2 c), respectively. The corresponding extremal graphs are also characterized. 2000 Mathematics Subject Classif...

Let G be a graph(directed or undirected) having k number of blocks. A B-partition of G is a partition into k vertex-disjoint subgraph (B̂1, B̂1, . . . , B̂k) such that B̂i is induced subgraph of Bi for i = 1, 2, . . . , k. The terms ∏k i=1 det(B̂i), ∏k i=1 per(B̂i) are det-summands and per-summands, respectively, corresponding to the B-partition. The determinant and permanent of a graph having no loo...

Let G be a connected graph with vertex set V (G). The degree resistance distance of G is defined as DR(G) = ∑ fu,vg V (G)[d(u) +d(v)]R(u, v), where d(u) is the degree of vertex u, and R(u, v) denotes the resistance distance between u and v. In this paper, we characterize n-vertex unicyclic graphs having minimum and second minimum degree resistance distance.

The class of connected 3-colored digraphs containing exactly one nonsingular cycle is considered in this article. The main objective is to study the smallest Laplacian eigenvalue and the corresponding eigenvectors of such graphs. It is shown that the smallest Laplacian eigenvalue of such a graph can be realized as the algebraic connectivity (second smallest Laplacian eigenvalue) of a suitable u...

For any simple graph G, let D(G) denote the degree set {degG(v) : v ∈ V (G)}. Let S be a finite, nonempty set of positive integers. In this paper, we first determine the families of graphs G which are unicyclic, bipartite satisfying D(G) = S, and further obtain the graphs of minimum orders in such families. More general, for a given pair (S, T ) of finite, nonempty sets of positive integers of ...

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...