On the inertia of weighted (k - 1)-cyclic graphs

Abstract

Let G w be a weighted graph. The inertia of G w is the triple In( G w ) = ( i + ( G w ), i − ( G w ), i 0 ( G w )) , where i + ( G w ), i − ( G w ), i 0 ( G w ) are, respectively, the number of the positive, negative and zero eigenvalues of the adjacency matrix A ( G w ) of G w including their multiplicities. A simple n -vertex connected graph is called a ( k − 1) -cyclic graph provided that its number of edges equals n + k − 2 . Let θ ( r 1 , r 2 , …, r k ) w be an n -vertex simple weighted graph obtained from k weighted paths ( P r 1 ) w , ( P r 2 ) w , …, ( P r k ) w by identifying their initial vertices and terminal vertices, respectively. Set Θ k : = { θ ( r 1 , r 2 , …, r k ) w : r 1 + r 2 + ⋯ + r k = n + 2 k − 2}. The inertia of the weighted graph θ ( r 1 , r 2 , …, r k ) w is studied. Also, the weighted ( k − 1) -cyclic graphs that contain θ ( r 1 , r 2 , …, r k ) w as an induced subgraph are studied. We characterize those graphs among Θ k that have extreme inertia. The results generalize the corresponding results obtained in [X.Z. Tan, B.L. Liu, The nullity of ( k − 1) -cyclic graphs, Linear Algebra Appl. 438 (2013) 3144-3153] and [G.H. Yu et al., The inertia of weighted unicyclic graphs, Linear Algebra Appl. 448 (2014) 130-152].