The harmonic index H(G) of a graph G is defined as the sum of weights 2 d(u) + d(v) of all edges uv of G, where d(u) denotes the degree of a vertex u in G. In this paper, we first present a sharp lower bound on the harmonic index of unicyclic conjugated graphs (unicyclic graphs with a perfect matching). Also a sharp lower bound on the harmonic index of unicyclic graphs is given in terms of the ...

The Harmonic index $ H(G) $ of a graph $ G $ is defined as the sum of the weights $ dfrac{2}{d(u)+d(v)} $ of all edges $ uv $ of $G$, where $d(u)$ denotes the degree of the vertex $u$ in $G$. In this work, we prove the conjecture $dfrac{H(G)}{D(G)} geq dfrac{1}{2}+dfrac{1}{3(n-1)}  $ given by Jianxi Liu in 2013 when G is a unicyclic graph and give a better bound $ dfrac{H(G)}{D(G)}geq dfra...

A set S ⊆ V is independent in a graph G = (V, E) if no two vertices from S are adjacent. The independence number α(G) is the cardinality of a maximum independent set, while μ(G) is the size of a maximum matching in G. If α(G) + μ(G) = |V |, then G is called a König-Egerváry graph. The number dc (G) = max{|A| − |N (A)| : A ⊆ V } is called the critical difference of G [21], where N (A) = {v : v ∈...

Let G be a simple connected graph and t be a given real number. The zero-order general Randić index αt(G) of G is defined as ∑ v∈V (G) d(v) t , where d(v) denotes the degree of v. In this paper, for any t , we characterize the graphs with the greatest and the smallest αt within two subclasses of connected unicyclic graphs on n vertices, namely, unicyclic graphs with k pendant vertices and unicy...

The Hosoya index Z(G) of a graph G is defined as the total number of edge independent sets of G. In this paper, we extend the research of [J. Ou, On extremal unicyclic molecular graphs with maximal Hosoya index, Discrete Appl. Math. 157 (2009) 391–397.] and [Y. Ye, X. Pan, H. Liu, Ordering unicyclic graphs with respect to Hosoya indices and Merrifield-Simmons indices, MATCH Commun. Math. Comput...

The resistance distance between two vertices of a graph can be defined as the effective resistance between the two vertices, when the graph is viewed as an electrical network with each edge carrying unit resistance. The concept has several different motivations. The resistance matrix of a graph is a matrix with its (i, j)-entry being the resistance distance between vertices i and j. We obtain a...

If G is a connected graph with vertex set V (G), then the eccentric connectivity index of G, denoted by ξc(G), is defined as ∑ v∈V (G) deg(v)ec(v), where deg(v) is the degree of a vertex v and ec(v) is its eccentricity. Morgan et al. [5] investigated the eccentric connectivity index of trees. In this paper, we investigate the eccentric connectivity index of unicyclic graphs. Upper bound is obta...

The detour index of a connected graph is defined as the sum of the detour distances (lengths of longest paths) between unordered pairs of vertices of the graph. We give bounds for the detour index, determine the graphs with minimum and maximum detour indices respectively in the class of n-vertex unicyclic graphs with cycle length r, where 3 ≤ r ≤ n−2, and the graphs with the first, the second a...

The purpose of this paper is to obtain a necessary and sufficient condition for the tensor product of two or more graphs to be connected, bipartite or eulerian. Also, we present a characterization of the duplicate graph $G 1 K_2$ to be unicyclic. Finally, the girth and the formula for computing the number of triangles in the tensor product of graphs are worked out.