The Turán type extremal problem asks to maximize the number of edges over all graphs which do not contain fixed subgraphs. Similarly, the spectral Turán type extremal problem asks to maximize spectral radius of all graphs which do not contain fixed subgraphs. In this paper, we determine the maximum spectral radius of all graphs without containing a linear forest as a subgraph and characterize all corresponding extremal graphs. In addition, the maximum number of edges and spectral radius of all bipartite graphs without containing k · P 3 as a subgraph are obtained and all extremal graphs are also characterized. Moreover, some relations between Tuán type extremal problems and spectral Turán type extremal problems are discussed.
Let Fa 1 ,...,a k be a graph consisting of k cycles of odd length 2a1 + 1, . . . , 2a k + 1, respectively which intersect in exactly a common vertex, where k ≥ 1 and a1 ≥ a2 ≥ • • • ≥ a k ≥ 1. In this paper, we present a sharp upper bound for the signless Laplacian spectral radius of all Fa 1 ,...,a k -free graphs and characterize all extremal graphs which attain the bound. The stability methods and structure of graphs associated with the eigenvalue are adapted for the proof.
Turán type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Turán type extremal problem is how to maximize (signless Laplacian) spectral radius over all graphs which do not contain fixed subgraphs. In this paper, we first present a stability result for k • P3 in terms of the number of edges and then determine all extremal graphs maximizing the signless Laplacian spectral radius over all graphs which do not contain a fixed linear forest with at most two odd paths or k•P3 as a subgraph, respectively.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.