On the largest eigenvalue of signed unicyclic graphs

The Hoffman program of graphs: old and new

“…Although Hoffman's theorem was ultimately based on a tree, its proof cannot be directly translated to S-spectra. In fact, if G is not a tree or a cycle, then ρ A (G) > ( √ 5 + 1)/2, whereas G <2 S contains signed unicyclic graphs which are not cycles and signed bicyclic graphs as well (see [2,4]. Problem 3.…”

On the index of unbalanced signed bicyclic graphs

“…It is not difficult to see that, we only need to consider the case that Γ ∈ {Γ 9 , Γ 10 , Γ i 2 , Γ j 3 } , where 1 ≤ i ≤ 7 and 1 ≤ j ≤ 5 ( as shown in Figure 5). By direct computation, we can prove that λ(Γ 5 ) > λ(Γ 9 ), λ(Γ 5 ) > λ(Γ 10 ), λ(Γ 5 ) > λ(Γ 1 2 ) = max{λ(Γ 1 2 ), . .…”

“…. , 5) be the unbalanced signed graphs as shown in Figure 4, then (1). the index λ(Γ i ) is the largest root of the equation f i (x) = 0, where (2).…”

“…One classical problem of graph spectra is to identify the extremal graphs with respect to the index in some given class of graphs. For signed graphs, since all signatures of a given tree are equivalent, the first non-trivial signature arises for unicyclic graphs, which was considered in [1]. The authors determined signed graphs achieving the minimal or the maximal index in the class of unbalanced unicyclic graphs of order n ≥ 3.…”

Ordering signed graphs with large index

“…B n ) denote the class of unbalanced unicyclic (resp. bicyclic) signed graphs of order n. Akbari et al [1] determined the signed graphs attaining the extremal indices in U n . Some of the same authors studied in [10] signed graphs achieving the maximum index among signed graphs in U n of fixed girth.…”