Spectral properties of a class of unicyclic graphs

Spectral properties of a class of unicyclic graphs

The eigenvalues of G are denoted by [Formula: see text], where n is the order of G. In particular, [Formula: see text] is called the spectral radius of G, [Formula: see text] is the least eigenvalue of G, and the spread of G is defined to be the difference between [Formula: see text] and [Formula: see text]. Let [Formula: see text] be the set of n-vertex unicyclic graphs, each of whose vertices...

The Laplacian Spectral Radius of a Class of Unicyclic Graphs

Let C(n, k) be the set of all unicyclic graphs with n vertices and cycle length k. For anyU ∈ C(n, k),U consists of the (unique) cycle (say Ck) of length k and a certain number of trees attached to the vertices of Ck having (in total) n − k edges. If there are at most two trees attached to the vertices of Ck, where k is even, we identify in the class of unicyclic graphs those graphs whose Lapla...

Order unicyclic mixed graphs by spectral radius

Detour Index of a Class of Unicyclic Graphs

The detour index of a connected graph is defined as the sum of detour distances between all unordered pairs of vertices. We determine the n-vertex unicyclic graphs whose vertices on its unique cycle all have degree at least three with the first, the second and the third smallest and largest detour indices respectively for n ≥ 7.

A class of unicyclic graphs determined by their Laplacian spectrum

Let Gr,p be a graph obtained from a path by adjoining a cycle Cr of length r to one end and the central vertex of a star Sp on p vertices to the other end. In this paper, it is proven that unicyclic graph Gr,p with r even is determined by its Laplacian spectrum except for n = p+4.