Tricyclic Graphs Research Articles

On the index of tricyclic graphs with perfect matchings

Let T ( 2 k ) be the set of all tricyclic graphs on 2 k ( k ⩾ 2 ) vertices with perfect matchings. In this paper, we discuss some properties of the connected graphs with perfect matchings, and then determine graphs with the largest index in T ( 2 k ) .

On a conjecture of the Randić index

The Randić index of a graph G is defined as R ( G ) = ∑ u ∼ v ( d ( u ) d ( v ) ) − 1 2 , where d ( u ) is the degree of vertex u and the summation goes over all pairs of adjacent vertices u , v . A conjecture on R ( G ) for connected graph G is as follows: R ( G ) ≥ r ( G ) − 1 , where r ( G ) denotes the radius of G . We proved that the conjecture is true for biregular graphs, connected graphs with order n ≤ 10 and tricyclic graphs.

On tricyclic graphs of a given diameter with minimal energy

The energy of a graph is defined as the sum of the absolute values of all the eigenvalues of the graph. Let G ( n , d ) be the class of tricyclic graphs G on n vertices with diameter d and containing no vertex disjoint odd cycles C p , C q of lengths p and q with p + q ≡ 2 ( mod 4 ) . In this paper, we characterize the graphs with minimal energy in G ( n , d ) .

On the nullity of graphs with pendent vertices

The nullity of a graph G, denoted by η ( G ) , is the multiplicity of the eigenvalue zero in its spectrum. Cheng and Liu [B. Cheng, B. Liu, On the nullity of graphs, Electron. J. Linear Algebra 16 (2007) 60–67] characterized the extremal graphs attaining the upper bound n - 2 and the second upper bound n - 3 . In this paper, as the continuance of it, we determine the extremal graphs with pendent vertices achieving the third upper bound n - 4 and fourth upper bound n - 5 . We then proceed recursively to construct all graphs with pendent vertices which satisfy η ( G ) > 0 . Our results provide a unified approach to determine n-vertex unicyclic (respectively, bicyclic and tricyclic) graphs which achieve the maximal and second maximal nullity and characterize n-vertex extremal trees attaining the second and third maximal nullity. As a consequence we, respectively, determine the nullity sets of trees, unicyclic graphs, bicyclic graphs and tricyclic graphs on n vertices.

The spectral radius of tricyclic graphs with n vertices and k pendent vertices

In this paper, we determine graphs with the largest spectral radius among all the tricyclic graphs with n vertices and k pendent vertices.

Tricylic hamiltonian graphs with minimal index

The index of a graph is the largest eigenvalue of an adjacency matrix whose entries are the real numbers 0 and 1. Among the tricyclic Hamiltonian graphs with a prescribed number of vertices, those graphs with minimal index are determined.