Graph defined by Birkhoff–James orthogonality relation in normed spaces is studied. It is shown that (i) in a normed space of sufficiently large dimension there always exists a nonzero vector which ...is mutually Birkhoff–James orthogonal to each among a fixed number of given vectors, and (ii) in nonsmooth norms the cardinality of the set of pairwise Birkhoff–James orthogonal vectors may exceed the dimension of the vector space, but this cardinality is always bounded above by a function of the dimension. It is further shown that any given pair of elements in a normed space can be extended to a finite tuple such that each consecutive elements are mutually Birkhoff–James orthogonal; the exact minimal length of the tuple is also determined.
Signed spectral Turań type theorems Rajesh Kannan, M.; Pragada, Shivaramakrishna
Linear algebra and its applications,
04/2023, Volume:
663
Journal Article
Peer reviewed
Open access
A signed graph Σ=(G,σ) is a graph where the function σ assigns either 1 or −1 to each edge of the simple graph G. The adjacency matrix of Σ, denoted by A(Σ), is defined canonically. In a recent ...paper, Wang et al. extended the spectral bounds of Hoffman and Cvetković for the chromatic number of signed graphs. They proposed an open problem related to the balanced clique number and the largest eigenvalue of a signed graph. We solve a strengthened version of this open problem. As a byproduct, we give alternate proofs for some of the known classical bounds for the least eigenvalues of unsigned graphs. We extend the Turán's inequality for the signed graphs. Besides, we study the Bollobás and Nikiforov conjecture for the signed graphs and show that the conjecture need not be true for the signed graphs. Nevertheless, the conjecture holds for signed graphs under some assumptions. Finally, we study some of the relationships between the number of signed walks and the largest eigenvalue of a signed graph.
The signless Laplacian spectral radius of a graph is the largest eigenvalue of its signless Laplacian matrix. In the paper, the graph with maximal signless Laplacian spectral radius among all graphs ...with given size and clique number is characterized. As applications, we determine the graph with maximal signless Laplacian spectral radius among all graphs with given size, and characterize the graph with maximal signless Laplacian spectral radius among all graphs with given size and chromatic number.
Zykov showed in 1949 that among graphs on n vertices with clique number ω(G)≤ω, the Turán graph Tω(n) maximizes not only the number of edges but also the number of copies of Kt for each size t. The ...problem of maximizing the number of copies of Kt has also been studied within other classes of graphs, such as those on n vertices with maximum degree Δ(G)≤Δ.
We combine these restrictions and investigate which graphs with Δ(G)≤Δ and ω(G)≤ω maximize the number of copies of Kt per vertex. We define ft(Δ,ω) as the supremum of ρt, the number of copies of Kt per vertex, among such graphs, and show for fixed t and ω that ft(Δ,ω)=(1+o(1))ρt(Tω(Δ+⌊Δω−1⌋)). For two infinite families of pairs (Δ,ω), we determine ft(Δ,ω) exactly for all t≥3. For another we determine ft(Δ,ω) exactly for the two largest possible clique sizes. Finally, we demonstrate that not every pair (Δ,ω) has an extremal graph that simultaneously maximizes the number of copies of Kt per vertex for every size t.
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