### 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*_{r1})_{w}, (*P*_{r2})_{w}, …, (*P*_{rk})_{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].

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DOI: https://doi.org/10.26493/1855-3974.673.4a1

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications