Unified approach for spectral properties of weighted adjacency matrices for graphs with degree-based edge-weights. (English) Zbl 1542.05104
Summary: Let \(G\) be a graph and \(d_i\) be the degree of a vertex \(v_i\) in \(G\). For a symmetric real function \(f(x, y)\), one can get an edge-weighted graph in such a way that for each edge \(v_i v_j\) of \(G\), the weight of \(v_i v_j\) is assigned by \(f(d_i, d_j)\). Hence, we have a weighted adjacency matrix \(A_f (G)\) of \(G\), in which the \(ij\)-entry is equal to \(f(d_i, d_j)\) if \(v_i v_j \in E(G)\) and 0 otherwise. In this paper, we use a unified approach to deal with the spectral properties of \(A_f (G)\) for \(f(x, y)\) to be the functions of graphical or topological function-indices. Firstly, we obtain uniform interlacing inequalities for the weighted adjacency eigenvalues. For the edge-weight functions defined by almost a half of popularly used topological indices, it can be shown that our inequalities cannot be improved. Secondly, we establish a uniform equivalent condition for a connected graph \(G\) to have \(m\) distinct weighted adjacency eigenvalues. As an application, a combinatorial characterization for a graph to have two and three distinct weighted adjacency eigenvalues is presented, respectively. Moreover, bipartite graphs and unicyclic graphs with three distinct weighted adjacency eigenvalues are characterized. This paper attempts to unify the spectral study for weighted adjacency matrices of graphs with degree-based edge-weights.
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C92 | Chemical graph theory |
| 05C09 | Graphical indices (Wiener index, Zagreb index, Randić index, etc.) |
Keywords:
degree-based edge-weight; weighted adjacency matrix (eigenvalue); topological function index; interlacing inequalityReferences:
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