A note on almost regular matrices

Walks and the spectral radius of graphs

“…Setting p = 2, r = 1, we obtain the following inequalities proved by Favaron, Mahéo, and Saclé [8], and in a wider context also by Hoffman, Wolfe, and Hofmeister [11],…”

“…As shown in [8] and [11] equality holds in ( 12) and ( 13) iff G is regular or semiregular. The case of equality in (10) and ( 11) is an open question.…”

Two shorter proofs in spectral graph theory

“…Since F. Runge had not explicitly required the graph to be connected, M. Hofmeister [5] weakened this to the conjecture that the condition (3) is sufficient for a graph to be almost regular. This conjecture was proved in [4], and here we give a new and shorter proof.…”

Bounds for Aα-eigenvalues

“…Remark 3.7. From Theorem 3.5 we have that the equality in (15) occurs if and only if 𝐺 is regular. Taking this extremal graph and applying its information in (10) we obtain…”

“…Let 𝐺 ∼ = 𝐾 1,6 as in Figure 2. The Table 2 presents a comparison of the upper bounds (10) and (15) and concludes that they are incomparable. Remark 3.7.…”