A wheel graph is a formed by connecting single vertex to all vertices of cycle. called wheel-free if it does not contain any as subgraph. In 2010, Nikiforov proposed Brualdi–Solheid–Turán type problem: what the maximum spectral radius order n that subgraphs particular kind. this paper, we study problem for graphs, and determine (signless Laplacian) n. Furthermore, characterize extremal graphs.

Introduction/purpose: Vertex-degree-based (VDB) graph matrices form a special class of matrices, corresponding to the currently much investigated vertex-degree-based invariants. Some spectral properties these are investigated. Results: Generally valid sharp lower and upper bounds established for radius any VDB matrix. The equality cases characterized. Several earlier published results shown be ...

Rigidity is the property of a structure that does not flex under an applied force. In past several decades, rigidity graphs has been widely studied in discrete geometry and combinatorics. Laman (1970) obtained combinatorial characterization rigid $\mathbb{R}^2$. Lovász Yemini (1982) proved every $6$-connected graph Jackson Jordán (2005) strengthened this result, showed globally Thus with algebr...

We show a simple method for constructing larger dimension nonnegative matrices with somewhat arbitrary entries which can be irreducible or reducible but preserving the spectral radius via row sum expansions. This yields sufficient criteria two square of to have same radius, way compare radii matrices, and derive new upper lower bounds on give standard as special case.

Let G be a simple connected graph. The terminal distance matrix of G is the distance matrix between all pendant vertices of G. In this paper, we introduce some general transformations that increase the terminal distance spectral radius of a connected graph and characterize the extremal trees with respect to the terminal distance spectral radius among all trees with a fixed number of pendant ver...

The Laplacian-energy-like of a simple connected graph G is defined as LEL:=LEL(G)=∑_(i=1)^n√(μ_i ), Where μ_1 (G)≥μ_2 (G)≥⋯≥μ_n (G)=0 are the Laplacian eigenvalues of the graph G. Some upper and lower bounds for LEL are presented in this note. Moreover, throughout this work, some results related to lower bound of spectral radius of graph are obtained using the term of ΔG as the num...

The sign-real and the sign-complex spectral radius, also called the generalized spectral radius, proved to be an interesting generalization of the classical Perron-Frobenius theory (for nonnegative matrices) to general real and to general complex matrices, respectively. Especially the generalization of the well-known Collatz-Wielandt max-min characterization shows one of the many one-to-one cor...

Replace certain edges of a directed graph by chains and consider the e ect on the spectrum of the graph. It is shown that the spectral radius decreases monotonically with the expansion and that, for a strongly connected graph that is not a single cycle, the spectral radius decreases strictly monotonically with the expansion. A limiting formula is given for the spectral radius of the expanded gr...

The sign-real and the sign-complex spectral radius, also called the generalized spectral radius, proved to be an interesting generalization of the classical Perron-Frobenius theory (for nonnegative matrices) to general real and to general complex matrices, respectively. Especially the generalization of the well-known Collatz-Wielandt max-min characterization shows one of the many one-to-one cor...

Let −→ G be a digraph and A( −→ G) be the adjacency matrix of −→ G . Let D( −→ G) be the diagonal matrix with outdegrees of vertices of −→ G and Q( −→ G) = D( −→ G) + A( −→ G) be the signless Laplacian matrix of −→ G . The spectral radius of Q( −→ G) is called the signless Laplacian spectral radius of −→ G . In this paper, we determine the unique digraph which attains the maximum (or minimum) s...