Graphs are used to model chemical compounds and drugs. In the graphs, each vertex represents an atom of molecule and edges between the corresponding vertices are used to represent covalent bounds between atoms. The Wiener polarity index W p (G) of a graph G is the number of unordered pairs of vertices u, v of G such that the distance between u and v is equal to 3. The trees and unicyclic graphs...

Let U(n, k) be the set of non-bipartite unicyclic graphs with n vertices and k pendant vertices, where n ≥ 4. In this paper, the unique graph with the minimal least eigenvalue of the signless Laplacian among all graphs in U(n, k) is determined. Furthermore, it is proved that the minimal least eigenvalue of the signless Laplacian is an increasing function in k. Let Un denote the set of non-bipar...

Let $$\Gamma =(K_{n},H^-)$$ be a signed complete graph whose negative edges induce subgraph H. The index of $$\Gamma$$ is the largest eigenvalue its adjacency matrix. In this paper, we study when H unicyclic graph. We show that among all graphs order $$n>5$$ k and maximizes index, triangle with remaining vertices being pendant at same vertex triangle.

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 of the respective graph. In this paper we present the lower bound on M1 and M2 among all unicyclic graphs of given order, maximum degree, and cycle length, and characterize graphs for which th...

The reverse degree distance of a connected graph $G$ is defined in discrete mathematical chemistry as [ r (G)=2(n-1)md-sum_{uin V(G)}d_G(u)D_G(u), ] where $n$, $m$ and $d$ are the number of vertices, the number of edges and the diameter of $G$, respectively, $d_G(u)$ is the degree of vertex $u$, $D_G(u)$ is the sum of distance between vertex $u$ and all other vertices of $G$, and $V(G)$ is the...

Let G be a connected graph on n vertices and D(G) its distance matrix. The formula for computing the determinant of this matrix in terms number is known when either tree or unicyclic graph. In work we generalize these results, obtaining any whose block decomposition consists edges, bicyclic graphs.

A k-partition Π = {S1,S2, . . . ,Sk} of V (G) is resolving if for every two distinct vertices u and v of a connected graph G, there is a set Si in Π so that the minimum distance between u and a vertex of Si is different from the minimum distance between v and a vertex of Si. A resolving partition Π is said to be connected if each subgraph < Si > induced by Si (1 ≤ i ≤ k) is connected in G. In t...

Let G be a simple connected graph with pendant vertex set ∂V and nonpendant vertex set V0. The signless Laplacian matrix of G is denoted by Q(G). The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ̸= 0 on V (G) such that Q(G)f(u) = λf(u) for u ∈ V0 and f(u) = 0 for u ∈ ∂V . The signless Dirichlet spectral radius λ(G) is the largest signless Dirichlet eigenva...

Let α (G) denote the maximum size of an independent set of vertices and μ (G) be the cardinality of a maximum matching in a graph G. A matching saturating all the vertices is a perfect matching. If α (G) + μ (G) = |V (G)|, then G is called a König-Egerváry graph. A graph is unicyclic if it has a unique cycle. It is known that a maximum matching can be found in O(m •√n) time for a graph with n v...

Several matrices can be associated to a graph such as the adjacency matrix or the Laplacian matrix. The spectrum of these matrices gives some informations about the structure of the graph and the question “Which graphs are determined by their spectrum?” remains a difficult problem in algebraic graph theory. In this article we enlarge the known families of graphs determined by their spectrum by ...