Extreme Sombor Spectral Radius 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...

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...

On the spectral radius of unicyclic graphs with fixed girth

The spectral radius of a graph is the largest eigenvalue of its adjacency matrix. Let U g n be the set of unicyclic graphs of order n with girth g. For all integers n and g with 5 ≤ g ≤ n − 6, we determine the first ⌊ g2⌋+ 3 spectral radii of unicyclic graphs in the set U g n .

The Signless Dirichlet Spectral Radius of Unicyclic Graphs

Let G be a simple connected graph with pendant vertex set ∂V and nonpendant vertex set V0. The signless Laplacian matrix of G is denoted by Q(G). The signless Dirichlet eigenvalue is a real number λ such that there exists a function f ̸= 0 on V (G) such that Q(G)f(u) = λf(u) for u ∈ V0 and f(u) = 0 for u ∈ ∂V . The signless Dirichlet spectral radius λ(G) is the largest signless Dirichlet eigenva...