Extremal Bicyclic Graph with Perfect Matching for Different Indices
نویسندگان
چکیده
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
منابع مشابه
The Harmonic Index for Unicyclic and Bicyclic Graphs with given Matching Number
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...
متن کاملOn Sum–Connectivity Index of Bicyclic Graphs
We determine the minimum sum–connectivity index of bicyclic graphs with n vertices and matching number m, where 2 ≤ m ≤ ⌊n2 ⌋, the minimum and the second minimum, as well as the maximum and the second maximum sum–connectivity indices of bicyclic graphs with n ≥ 5 vertices. The extremal graphs are characterized. MSC 2000: 05C90; 05C35; 05C07
متن کاملOn the harmonic index of bicyclic 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...
متن کاملPerfect Matchings in Edge-Transitive Graphs
We find recursive formulae for the number of perfect matchings in a graph G by splitting G into subgraphs H and Q. We use these formulas to count perfect matching of P hypercube Qn. We also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in G, is the graph constructed from by deleting edges with an en...
متن کاملSharp upper bounds on Zagreb indices of bicyclic graphs with a given matching number
For a (molecular) graph, the first Zagreb index M1 is equal to the sum of squares of the vertex degrees, and the second Zagreb index M2 is equal to the sum of products of degrees of pairs of adjacent vertices. In this paper, we investigate Zagreb indices of bicyclic graphs with a given matching number. Sharp upper bounds for the first and second Zagreb indices of bicyclic graphs in terms of the...
متن کامل