In 1973, Schwenk proved that almost all trees are cospectral, showing that the identification of families of trees without cospectral mates among trees is a challenging problem. In this paper, we study the family of uniform starlike type trees Skp, defined as those trees S for which S−v is the disjoint union of p copies of the path Pk. We prove that every S∈Skp is determined by the adjacency spectrum among connected graphs for p≥3, and we provide a complete spectral characterization of the family S2p among all graphs.
By the second maximum multiplicity eigenvalue for a graph G, we mean the second largest multiplicity eigenvalue among real symmetric matrices with a given graph G and that also have the maximum multiplicity eigenvalue for that graph. Described here is a purely combinatorial characterization of the second maximum multiplicity for a tree. It uses elements of the classical description of the maximum multiplicity, but proofs are more complicated. An important by-product of this work is a purely combinatorial determination, for each vertex of a tree, those that may be Parter for the path cover number, that must be downer, and that must be neutral. These are the only possibilities. Examples are given.
For positive integers n and t, let Γ(n,2,t) be the family of n×n 0–1 matrices A for which every entry of A2 is at most t, and let γ(n,2,t) be the largest possible number of ones in such a matrix. A previous result established the exact value of γ(n,2,t) under a cubic lower bound on n. We reduce this sufficient order condition to a quadratic one. More precisely, for t≥2, the known extremal formula holds when n−t is even and n≥2t2+6t, and when n−t is odd and n≥4t2+9t+2. We also obtain an exchange-rigidity property for extremal matrices. The proof retains the exact deficit associated with the gap between the maximum degree and its extremal value.
In this paper, we study unitary associative G-graded algebras endowed with a graded involution with polynomial codimension growth. For each fixed k≥1, we construct algebras whose codimension sequences grow polynomially of degree k, realizing the maximal and the minimal leading coefficients. We then provide, up to equivalence, a complete classification of finitely generated unitary algebras with quadratic codimension growth. More precisely, we establish a direct correspondence between the nonzero multiplicities appearing in the decomposition of the proper cocharacters and the algebras generating minimal varieties within the given variety. As a consequence, every affine unitary variety with at most quadratic growth is generated by an algebra that decomposes into a direct sum of algebras generating minimal varieties of at most quadratic growth.
