The spectral classes of unicyclic graphs

Order unicyclic mixed graphs by spectral radius

New upper bounds on the spectral radius of unicyclic graphs

Let G = (V (G), E(G)) be a unicyclic simple undirected graph with largest vertex degree . Let Cr be the unique cycle of G. The graph G− E(Cr ) is a forest of r rooted trees T1,T2, . . .,Tr with root vertices v1, v2, . . ., vr , respectively. Let k(G) = max 1 i r {max{dist(vi , u) : u ∈ V (Ti )}} + 1, where dist(v, u) is the distance from v to u. Let μ1(G) and λ1(G) be the spectral radius of the...

Total Vertex Irregularity Strength of Certain Classes of Unicyclic Graphs *

A total vertex irregular k-labeling φ of a graph G is a labeling of the vertices and edges of G with labels from the set {1, 2, . . . , k} in such a way that for any two different vertices x and y their weights wt(x) and wt(y) are distinct. Here, the weight of a vertex x in G is the sum of the label of x and the labels of all edges incident with the vertex x. The minimum k for which the graph G...

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...

Ordering of the signless Laplacian spectral radii of unicyclic graphs

For n ≥ 11, we determine all the unicyclic graphs on n vertices whose signless Laplacian spectral radius is at least n− 2. There are exactly sixteen such graphs and they are ordered according to their signless Laplacian spectral radii.