Nowhere-zero Flows Of Product Of Graphs And Eigenvalue Of Unicyclic Graphs With Perfect Matchings

Suppose that G = (V(G), E(G)) is a graph with vertex set V(G) = {v₁,…,v_n} and edge set E(G) = {e₁,…,e_n}. Let k be a positive integer. A nowhere-zero k-flow of G (abbreviated as k-NZF) is an ordered pair(D, f), where D is an orientation of G, and f : E(G) → {±1, ±2…,±(k-1)}is a function on E(G) such that Σ_e∈E+(v)f(e)=Σ_e∈E-(v)f(e) holds for every vertex v ∈V(G), where E⁺(v) and E^-(v) denote the set of outgoing and incoming edges incident with v with respect to D, respectively.We call n×n matrix A(G) = (a_ij), where a_ij = 1 if v_i is adjacent to v_j, and a_ij= 0 otherwise, the adjacent matrix of G. The characteristic polynomial of G is P(G;λ) = det(λI - A(G)). The roots of P(G; λ) are also called the eigenvalues of G.First, we characterize graphs whose tensor product and lexicographic product admit nowhere-zero 3-flows. The main results are: For two graphs G₁ and G₂ with δ_G1 ≥ 2 and G₂ not belonging to a well-characterized class of graphs, the tensor product of G₁ and G₂ admits a nowhere-zero 3-flow. For a graph G₁ composed of nontrivial components and a graph G₂ of order at least 3, the lexicographic product of G₁ and G₂ admits a nowhere-zero 3-flow.Second, an upper bound for the second largest eigenvalue of a unicyclic graphs with perfect matchings is obtained. At the same time, a lower bound for maximum value of the second largest eigenvalue of unicyclic graphs with perfecting matchings is also given. We also note that our bound is asymoptot-ically good.