tricyclic graph

‎Consider a simple‎, ‎undirected graph $ G=(V,E)$‎, ‎where $A$ represents the adjacency matrix and $Q$ represents the Laplacian matrix of $G$‎. ‎The second smallest eigenvalue of Laplacian matrix of $G$ is called the algebraic connectivity of $G$‎. ‎In this article‎, ‎we present a Python program for studying the Laplacian eigenvalues of a graph‎. ‎Then‎, ‎we determine the unique graph of minimum algebraic connectivity in the set of all tricyclic graphs‎.

Let $G$ be a connected graph on $n$ vertices. $G$ is called tricyclic if it has $n 2$ edges, and tetracyclic if $G$ has exactly $n 3$ edges. Suppose $mathcal{C}_n$ and $mathcal{D}_n$ denote the set of all tricyclic and tetracyclic $n-$vertex graphs, respectively. The aim of this paper is to calculate the minimum and maximum of eccentric connectivity index in $mathcal{C}_n$ and $mathcal{D}_n$.

Let dn;m = 2n+1 and En;m be the graph obtained from a path Pdn;m+1 = v0v1:::vdn;m by joining each vertex of Kn by joining each vertex of Kn. Zhang, Liu and Zhou [On the maximal eccentric connectivity indices of graphs, Appl. Math. J. Chinese Univ., in press] conjectured that if dn;m > 3, then En;m is the graph with maximal eccentric connectivity ndex among all connected graph with n vertices and m edges. In this note, we prove this conjecture. Moreover, we present the graph with maximal eccentric connectivity index among the connected graphs with n vertices. Finally, the minimum of this graph invariant n the classes of tricyclic and tetracyclic graphs are computed.