Given a molecular graph G, the Hosoya index Z(G) of G is defined as the total number of the matchings of the graph. Let Bn denote the set of bicyclic graphs on n vertices. In this paper, the minimal, the second-, the third-, the fourth-, and the fifth-minimal Hosoya indices of bicyclic graphs in the set Bn are characterized.

Let B(2m,m) be the set of all bicyclic graphs on 2m(m ≥ 2) vertices with perfect matchings. In this paper, we characterize the bicyclic graphs with minimal number of matchings and maximal number of independent sets in B(2m,m). 2010 Mathematics Subject Classification: 05C69, 05C05

In a certain class of graphs, a graph is called minimizing if the least eigenvalue of its adjacency matrix attains the minimum. A connected graph containing two or three cycles is called a bicyclic graph if its number of edges is equal to its number of vertices plus one. Let Gc 1,n and Gc 2,n be the classes of the connected graphs of order n whose complements are bicyclic with exactly two and t...

Let U(n, g) and B(n, g) be the set of unicyclic graphs and bicyclic graphs on n vertices with girth g, respectively. Let B1(n, g) be the subclass of B(n, g) consisting of all bicyclic graphs with two edge-disjoint cycles and B2(n, g) = B(n, g)\B1(n, g). This paper determines the unique graph with the maximal signless Laplacian spectral radius among all graphs in U(n, g) and B(n, g), respectivel...

In this paper, we study the signless Laplacian spectral radius of bicyclic graphs with given number of pendant vertices and characterize the extremal graphs.

The harmonic index of a graph $G$, denoted by $H(G)$, is defined asthe sum of weights $2/[d(u)+d(v)]$ over all edges $uv$ of $G$, where$d(u)$ denotes the degree of a vertex $u$. Hu and Zhou [Y. Hu and X. Zhou, WSEAS Trans. Math. {bf 12} (2013) 716--726] proved that for any bicyclic graph $G$ of order $ngeq 4$, $H(G)le frac{n}{2}-frac{1}{15}$ and characterize all extremal bicyclic graphs.In this...

The distance d(va,vb) between two vertices of a simple connected graph G is the length shortest path va and vb. Vertices va,vb are considered to be resolved by vertex v if d(va,v)≠d(vb,v). An ordered set W={v1,v2,v3,…,vs}⊆V(G) said resolving for G, any va,vb∈V(G),∃vi∈W∋d(va,vi)≠d(vb,vi). representation with respect W denoted r(v|W) an s-vector(s-tuple) (d(v,v1),d(v,v2),d(v,v3),…,d(v,vs)). Using...

It is well known that the graph invariant, ‘the Merrifield–Simmons index’ is important one in structural chemistry. The connected acyclic graphs with maximal and minimal Merrifield–Simmons indices are determined by Prodinger and Tichy [H. Prodinger, R.F. Tichy, Fibonacci numbers of graphs, Fibonacci Quart. 20 (1982) 16–21]. The sharp upper and lower bounds for theMerrifield–Simmons indices of u...