Abstract The graph-parameters mainly studies vertex degree and distance between unordered vertices. There are many graph-parameters, this paper focuses on the Gutman index. For sake of discussion, puts forward concept “ contribution” based relationship index graph vertex-Gutman On basis, we discussed unicyclic with pendent edges in odd even parts, gave corresponding extremal graphs, which lays ...

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...

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.

In this paper, we give graph-theoretic algorithms of linear time to the Minimum All-Ones Problem for unicyclic and bicyclic graphs. These algorithms are based on a graph-theoretic algorithm of linear time to the Minimum All-Ones Problem with Restrictions for trees.

Denote by $\mathcal{G}(n,c,g,k)$ the set of all connected graphs order $n$, having $c$ cycles, girth $g$, and $k$ pendant vertices. In this paper, we give a partial characterisation structure those in maximising number induced subgraphs. For special case where $c=1$, find complete maximal unicyclic graphs. We also derive precise formula for corresponding maximum given following parameters: (1) ...

By d(v|G) and d_2(v|G) are denoted the number of first and second neighborsof the vertex v of the graph G. The first, second, and third leap Zagreb indicesof G are defined asLM_1(G) = sum_{v in V(G)} d_2(v|G)^2, LM_2(G) = sum_{uv in E(G)} d_2(u|G) d_2(v|G),and LM_3(G) = sum_{v in V(G)} d(v|G) d_2(v|G), respectively. In this paper, we generalizethe results of Naji et al. [Commun. Combin. Optim. ...

A total global dominator coloring of a graph is proper vertex with respect to which every in dominates color class, not containing and does dominate another class. The minimum number colors required such called the chromatic number, denoted by . In this paper, trees unicyclic graphs are explored.