On the eigenvalues and spectral radius of starlike trees

Let \(k\ge 1\) and \(n_1,\ldots ,n_k\ge 1\) be some integers. Let \(S(n_1,\ldots ,n_k)\) be a tree T such that T has a vertex v of degree k and \(T{\setminus } v\) is the disjoint union of the paths \(P_{n_1},\ldots ,P_{n_k}\), that is \(T{\setminus } v\cong P_{n_1}\cup \cdots \cup P_{n_k}\) so that every neighbor of v in T has degree one or two. The tree \(S(n_1,\ldots ,n_k)\) is called starlike tree, a tree with exactly one vertex of degree greater than two, if \(k\ge 3\). In this paper we obtain the eigenvalues of starlike trees. We find some bounds for the largest eigenvalue (for the spectral radius) of starlike trees. In particular we prove that if \(k\ge 4\) and \(n_1,\ldots ,n_k\ge 2\), then \(\frac{k-1}{\sqrt{k-2}}<\lambda _1(S(n_1,\ldots ,n_k))<\frac{k}{\sqrt{k-1}}\), where \(\lambda _1(T)\) is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval \((-2,2)\).     

Coloring Complete and Complete Bipartite Graphs from Random Lists

Assign to each vertex v of the complete graph \(K_n\) on n vertices a list L(v) of colors by choosing each list independently and uniformly at random from all f(n)-subsets of a color set \([n] = \{1,\dots , n\}\), where f(n) is some integer-valued function of n. Such a list assignment L is called a random (f(n), [n])-list assignment. In this paper, we determine the asymptotic probability (as \(n \rightarrow \infty \)) of the existence of a proper coloring \(\varphi \) of \(K_n\), such that \(\varphi (v) \in L(v)\) for every vertex v of \(K_n\). We show that this property exhibits a sharp threshold at \(f(n) = \log n\). Additionally, we consider the corresponding problem for the line graph of a complete bipartite graph \(K_{m,n}\) with parts of size m and n, respectively. We show that if \(m = o(\sqrt{n})\), \(f(n) \ge 2 \log n\), and L is a random (f(n), [n])-list assignment for the line graph of \(K_{m,n}\), then with probability tending to 1, as \(n \rightarrow \infty \), there is a proper coloring of the line graph of \(K_{m,n}\) with colors from the lists.     

The Best Mixing Time for Random Walks on Trees

We characterize the extremal structures for mixing walks on trees that start from the most advantageous vertex. Let \(G=(V,E)\) be a tree with stationary distribution \(\pi \). For a vertex \(v \in V\), let \(H(v,\pi )\) denote the expected length of an optimal stopping rule from v to \(\pi \). The best mixing time for G is \(\min _{v \in V} H(v,\pi )\). We show that among all trees with \(|V|=n\), the best mixing time is minimized uniquely by the star. For even n, the best mixing time is maximized by the uniquely path. Surprising, for odd n, the best mixing time is maximized uniquely by a path of length \(n-1\) with a single leaf adjacent to one central vertex.     

On convex iterative roots of non-monotonic mappings

Let I be an interval. We consider the non-monotonic convex self-mappings \(f:I\rightarrow I\) such that \(f^2\) is convex. They have the property that all iterates \(f^n\) are convex. In the class of these mappings we study three families of functions possessing convex iterative roots. A function f is said to be iteratively convex if f possesses convex iterative roots of all orders. A mapping f is said to be dyadically convex if for every \(n\ge 2\) there exists a convex iterative root \(f^{1/2^n}\) of order \(2^n\) and the sequence \(\{f^{1/2^n}\}\) satisfies the condition of compatibility, that is \( f^{1/2^n}\circ f^{1/2^n}= f^{1/2^{n-1}}.\) A function f is said to be flowly convex if it possesses a convex semi-flow of f, that is a family of convex functions \(\{f^t,t>0\}\) such that \(f^t\circ f^s=f^{t+s}, \ \ t,s >0\) and \(f^1=f\). We show the relations among these three types of convexity and we determine all convex iterative roots of non-monotonic functions.     

An <Emphasis Type="Italic">A</Emphasis>-invariant subspace for bipartite distance-regular graphs with exactly two irreducible <Emphasis Type="Italic">T</Emphasis>-modules with endpoint 2, both thin

Let \(\Gamma \) denote a bipartite distance-regular graph with vertex set X, diameter \(D \ge 4\), and valency \(k \ge 3\). Let \({{\mathbb {C}}}^X\) denote the vector space over \({{\mathbb {C}}}\) consisting of column vectors with entries in \({{\mathbb {C}}}\) and rows indexed by X. For \(z \in X\), let \({{\widehat{z}}}\) denote the vector in \({{\mathbb {C}}}^X\) with a 1 in the z-coordinate, and 0 in all other coordinates. Fix a vertex x of \(\Gamma \) and let \(T = T(x)\) denote the corresponding Terwilliger algebra. Assume that up to isomorphism there exist exactly two irreducible T-modules with endpoint 2, and they both are thin. Fix \(y \in X\) such that \(\partial (x,y)=2\), where \(\partial \) denotes path-length distance. For \(0 \le i,j \le D\) define \(w_{ij}=\sum {{\widehat{z}}}\), where the sum is over all \(z \in X\) such that \(\partial (x,z)=i\) and \(\partial (y,z)=j\). We define \(W=\mathrm{span}\{w_{ij} \mid 0 \le i,j \le D\}\). In this paper we consider the space \(MW=\mathrm{span}\{mw \mid m \in M, w \in W\}\), where M is the Bose–Mesner algebra of \(\Gamma \). We observe that MW is the minimal A-invariant subspace of \({{\mathbb {C}}}^X\) which contains W, where A is the adjacency matrix of \(\Gamma \). We show that \(4D-6 \le \mathrm{dim}(MW) \le 4D-2\). We display a basis for MW for each of these five cases, and we give the action of A on these bases.     

Relative Difference Sets and Hadamard Matrices from Perfect Quaternionic Arrays

Let \(G=\mathbf{C}_{n_1}\times \cdots \times \mathbf{C}_{n_m}\) be an abelian group of order \(n=n_1\dots n_m\), where each \(\mathbf{C}_{n_t}\) is cyclic of order \(n_t\). We present a correspondence between the (4n, 2, 4n, 2n)-relative difference sets in \(G\times Q_8\) relative to the centre \(Z(Q_8)\) and the perfect arrays of size \(n_1\times \dots \times n_m\) over the quaternionic alphabet \(Q_8\cup qQ_8\), where \(q=(1+i+j+k)/2\). In view of this connection, for \(m=2\) we introduce new families of relative difference sets in \(G\times Q_8\), as well as new families of Williamson and Ito Hadamard matrices with G-invariant components.     

A Completion of the Spectrum for the Overlarge Sets of Pure Mendelsohn Triple Systems

A pure Mendelsohn triple system of order v, denoted by PMTS(v), is a pair \((X,\mathcal {B})\) where X is a v-set and \(\mathcal {B}\) is a collection of cyclic triples on X such that every ordered pair of X belongs to exactly one triple of \(\mathcal {B}\) and if \(\langle a,b,c\rangle \in \mathcal {B}\) implies \(\langle c,b,a\rangle \notin \mathcal {B}\). An overlarge set of PMTS(v), denoted by OLPMTS(v), is a collection \(\{(Y{\setminus }\{y_i\},{\mathcal {A}}_i)\}_i\), where Y is a \((v+1)\)-set, \(y_i\in Y\), each \((Y{\setminus }\{y_i\},{\mathcal {A}}_i)\) is a PMTS(v) and these \({\mathcal {A}}_i\)s form a partition of all cyclic triples on Y. It is shown in [3] that there exists an OLPMTS(v) for \(v\equiv 1,3\) (mod 6), \(v>3\), or \(v \equiv 0,4\) (mod 12). In this paper, we shall discuss the existence problem of OLPMTS(v)s for \(v\equiv 6,10\) (mod 12) and get the following conclusion: there exists an OLPMTS(v) if and only if \(v\equiv 0,1\) (mod 3), \(v>3\) and \(v\ne 6\).     

On Star–Wheel Ramsey Numbers

For two given graphs \(G_1\) and \(G_2\), the Ramsey number \(R(G_1,G_2)\) is the least integer r such that for every graph G on r vertices, either G contains a \(G_1\) or \(\overline{G}\) contains a \(G_2\). In this note, we determined the Ramsey number \(R(K_{1,n},W_m)\) for even m with \(n+2\le m\le 2n-2\), where \(W_m\) is the wheel on \(m+1\) vertices, i.e., the graph obtained from a cycle \(C_m\) by adding a vertex v adjacent to all vertices of the \(C_m\).     

The Terwilliger algebra of the incidence graph of the Hamming graph

Let \(\varGamma \) be a distance-semiregular graph on Y, and let \(D^Y\) be the diameter of \(\varGamma \) on Y. Let \(\varDelta \) be the halved graph of \(\varGamma \) on Y. Fix \(x \in Y\). Let T and \(T'\) be the Terwilliger algebras of \(\varGamma \) and \(\varDelta \) with respect to x, respectively. Assume, for an integer i with \(1 \le 2i \le D^Y\) and for \(y,z \in \varGamma _{2i}(x)\) with \(\partial _{\varGamma }(y,z)=2\), the numbers \(|\varGamma _{2i-1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) and \(|\varGamma _{2i+1}(x) \cap \varGamma (y) \cap \varGamma (z)|\) depend only on i and do not depend on the choice of y, z. The first goal in this paper is to show the relations between T-modules of \(\varGamma \) and \(T'\)-modules of \(\varDelta \). Assume \(\varGamma \) is the incidence graph of the Hamming graph H(D, n) on the vertex set Y and the set \({\mathcal {C}}\) of all maximal cliques. Then, \(\varGamma \) satisfies above assumption and \(\varDelta \) is isomorphic to H(D, n). The second goal is to determine the irreducible T-modules of \(\varGamma \). For each irreducible T-module W, we give a basis for W the action of the adjacency matrix on this basis and we calculate the multiplicity of W.     

Simultaneous power factorization in modules over Banach algebras

Let A be a Banach algebra with a bounded left approximate identity \(\{e_\lambda \}_{\lambda \in \Lambda }\), let \(\pi \) be a continuous representation of A on a Banach space X, and let S be a non-empty subset of X such that \(\lim _{\lambda }\pi (e_\lambda )s=s\) uniformly on S. If S is bounded, or if \(\{e_\lambda \}_{\lambda \in \Lambda }\) is commutative, then we show that there exist \(a\in A\) and maps \(x_n: S\rightarrow X\) for \(n\ge 1\) such that \(s=\pi (a^n)x_n(s)\) for all \(n\ge 1\) and \(s\in S\). The properties of \(a\in A\) and the maps \(x_n\), as produced by the constructive proof, are studied in some detail. The results generalize previous simultaneous factorization theorems as well as Allan and Sinclair’s power factorization theorem. In an ordered context, we also consider the existence of a positive factorization for a subset of the positive cone of an ordered Banach space that is a positive module over an ordered Banach algebra with a positive bounded left approximate identity. Such factorizations are not always possible. In certain cases, including those for positive modules over ordered Banach algebras of bounded functions, such positive factorizations exist, but the general picture is still unclear. Furthermore, simultaneous pointwise power factorizations for sets of bounded maps with values in a Banach module (such as sets of bounded convergent nets) are obtained. A worked example for the left regular representation of \(\mathrm {C}_0({\mathbb R})\) and unbounded S is included.     

Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers q, n, d, let \(A_q(n,d)\) denote the maximum cardinality of a code of length n over an alphabet [q] with q letters and with minimum distance at least d. We consider the following upper bound on \(A_q(n,d)\). For any k, let \(\mathcal{C}_k\) be the collection of codes of cardinality at most k. Then \(A_q(n,d)\) is at most the maximum value of \(\sum _{v\in [q]^n}x(\{v\})\), where x is a function \(\mathcal{C}_4\rightarrow {\mathbb {R}}_+\) such that \(x(\emptyset )=1\) and \(x(C)=\!0\) if C has minimum distance less than d, and such that the \(\mathcal{C}_2\times \mathcal{C}_2\) matrix \((x(C\cup C'))_{C,C'\in \mathcal{C}_2}\) is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in n. It yields the new upper bounds \(A_4(6,3)\le 176\), \(A_4(7,3)\le 596\), \(A_4(7,4)\le 155\), \(A_5(7,4)\le 489\), and \(A_5(7,5)\le 87\).     

Anti-Ramsey Numbers of Graphs with Small Connected Components

The anti-Ramsey number, AR(n, G), for a graph G and an integer \(n\ge |V(G)|\), is defined to be the minimal integer r such that in any edge-colouring of \(K_n\) by at least r colours there is a multicoloured copy of G, namely, a copy of G that each of its edges has a distinct colour. In this paper we determine, for large enough \(n,\, AR(n,L\cup tP_2)\) and \(AR(n,L\cup kP_3)\) for any large enough t and k, and a graph L satisfying some conditions. Consequently, we determine AR(n, G), for large enough n, where G is \(P_3\cup tP_2\) for any \(t\ge 3,\, P_4\cup tP_2\) and \(C_3\cup tP_2\) for any \(t\ge 2,\, kP_3\) for any \(k\ge 3,\, tP_2\cup kP_3\) for any \(t\ge 1,\, k\ge 2\), and \(P_{t+1}\cup kP_3\) for any \(t\ge 3,\, k\ge 1\). Furthermore, we obtain upper and lower bounds for AR(n, G), for large enough n, where G is \(P_{k+1}\cup tP_2\) and \(C_k\cup tP_2\) for any \(k\ge 4,\, t\ge 1\).     

A new series of large sets of subspace designs over the binary field

In this article, we show the existence of large sets \({\text {LS}}_2[3](2,k,v)\) for infinitely many values of k and v. The exact condition is \(v \ge 8\) and \(0 \le k \le v\) such that for the remainders \(\bar{v}\) and \(\bar{k}\) of v and k modulo 6 we have \(2 \le \bar{v} < \bar{k} \le 5\). The proof is constructive and consists of two parts. First, we give a computer construction for an \({\text {LS}}_2[3](2,4,8)\), which is a partition of the set of all 4-dimensional subspaces of an 8-dimensional vector space over the binary field into three disjoint 2-\((8, 4, 217)_2\) subspace designs. Together with the already known \({\text {LS}}_2[3](2,3,8)\), the application of a recursion method based on a decomposition of the Graßmannian into joins yields a construction for the claimed large sets.     

Packing chromatic number versus chromatic and clique number

The packing chromatic number \(\chi _{\rho }(G)\) of a graph G is the smallest integer k such that the vertex set of G can be partitioned into sets \(V_i\), \(i\in [k]\), where each \(V_i\) is an i-packing. In this paper, we investigate for a given triple (a, b, c) of positive integers whether there exists a graph G such that \(\omega (G) = a\), \(\chi (G) = b\), and \(\chi _{\rho }(G) = c\). If so, we say that (a, b, c) is realizable. It is proved that \(b=c\ge 3\) implies \(a=b\), and that triples \((2,k,k+1)\) and \((2,k,k+2)\) are not realizable as soon as \(k\ge 4\). Some of the obtained results are deduced from the bounds proved on the packing chromatic number of the Mycielskian. Moreover, a formula for the independence number of the Mycielskian is given. A lower bound on \(\chi _{\rho }(G)\) in terms of \(\Delta (G)\) and \(\alpha (G)\) is also proved.     

The non-abelian tensor square of residually finite groups

Let m, n be positive integers and p a prime. We denote by \(\nu (G)\) an extension of the non-abelian tensor square \(G \otimes G\) by \(G \times G\). We prove that if G is a residually finite group satisfying some non-trivial identity \(f \equiv ~1\) and for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) such that \([x,y^{\varphi }]^q = 1\), then the derived subgroup \(\nu (G)'\) is locally finite (Theorem A). Moreover, we show that if G is a residually finite group in which for every \(x,y \in G\) there exists a p-power \(q=q(x,y)\) dividing \(p^m\) such that \([x,y^{\varphi }]^q\) is left n-Engel, then the non-abelian tensor square \(G \otimes G\) is locally virtually nilpotent (Theorem B).     

Optimal channel assignment and <Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">p</Emphasis>, 1)-labeling

The optimal channel assignment is an important optimization problem with applications in optical networks. This problem was formulated to the L(p, 1)-labeling of graphs by Griggs and Yeh (SIAM J Discrete Math 5:586–595, 1992). A k-L(p, 1)-labeling of a graph G is a function \(f:V(G)\rightarrow \{0,1,2,\ldots ,k\}\) such that \(|f(u)-f(v)|\ge p\) if \(d(u,v)=1\) and \(|f(u)-f(v)|\ge 1\) if \(d(u,v)=2\), where d(u, v) is the distance between the two vertices u and v in the graph. Denote \(\lambda _{p,1}^l(G)= \min \{k \mid G\) has a list k-L(p, 1)-labeling\(\}\). In this paper we show upper bounds \(\lambda _{1,1}^l(G)\le \Delta +9\) and \(\lambda _{2,1}^l(G)\le \max \{\Delta +15,29\}\) for planar graphs G without 4- and 6-cycles, where \(\Delta \) is the maximum vertex degree of G. Our proofs are constructive, which can be turned to a labeling (channel assignment) method to reach the upper bounds.     

Derivations Vanishing Identities Involving Generalized Derivations and Multilinear Polynomial in Prime Rings

Let R be a prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, \(f(x_1,\ldots ,x_n)\) be a multilinear polynomial over C, which is not central valued on R. Suppose that d is a non-zero derivation of R, F and G are two generalized derivations of R such that \(d\{F(u)u-uG^2(u)\}=0\) for all \(u\in f(R)\). Then one of the following holds:

1. (i)
   there exist \(a, b, p\in U\), \(\lambda \in C\) such that \(F(x)=\lambda x+bx+xa^2\), \(G(x)=ax\), \(d(x)=[p, x]\) for all \(x\in R\) with \([p, b]=0\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R;
    
2. (ii)
   there exist \(a, b, p\in U\) such that \(F(x)=ax\), \(G(x)=xb\), \(d(x)=[p,x]\) for all \(x\in R\) and \(f(x_1,\ldots , x_n)^2\) is central valued on R with \([p, a-b^2]=0\);
    
3. (iii)
   there exist \(a\in U\) such that \(F(x)=xa^2\) and \(G(x)=ax\) for all \(x\in R\);
    
4. (iv)
   there exists \(a\in U\) such that \(F(x)=a^2x\) and \(G(x)=xa\) for all \(x\in R\) with \(a^2\in C\);
    
5. (v)
   there exist \(a, p\in U\), \(\lambda , \alpha , \mu \in C\) such that \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) and \(d(x)=[p,x]\) for all \(x\in R\) with \(a^2=\mu -\alpha p\) and \(\alpha p^2+(\lambda -2\mu ) p\in C\);
    
6. (vi)
   there exist \(a\in U\), \(\lambda \in C\) such that R satisfies \(s_4\) and either \(F(x)=\lambda x+xa^2\), \(G(x)=ax\) or \(F(x)=\lambda x-a^2x\), \(G(x)=xa\) for all \(x\in R\).
    

     

Sets of minimal distances and characterizations of class groups of Krull monoids

Let H be a Krull monoid with finite class group G such that every class contains a prime divisor. Then every non-unit \(a \in H\) can be written as a finite product of atoms, say \(a=u_1 \cdot \ldots \cdot u_k\). The set \(\mathsf L (a)\) of all possible factorization lengths k is called the set of lengths of a. There is a constant \(M \in \mathbb N\) such that all sets of lengths are almost arithmetical multiprogressions with bound M and with difference \(d \in \Delta ^* (H)\), where \(\Delta ^* (H)\) denotes the set of minimal distances of H. We study the structure of \(\Delta ^* (H)\) and establish a characterization when \(\Delta ^*(H)\) is an interval. The system \(\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}\) of all sets of lengths depends only on the class group G, and a standing conjecture states that conversely the system \(\mathcal L (H)\) is characteristic for the class group. We confirm this conjecture (among others) if the class group is isomorphic to \(C_n^r\) with \(r,n \in \mathbb N\) and \(\Delta ^*(H)\) is not an interval.     

A Compactness Theorem of the Space of Free Boundary <Emphasis Type="Italic">f</Emphasis>-Minimal Surfaces in Three-Dimensional Smooth Metric Measure Space with Boundary

Let \((M^3,g,e^{-f}d\mu _M)\) be a compact three-dimensional smooth metric measure space with nonempty boundary. Suppose that M has nonnegative Bakry–Émery Ricci curvature and the boundary \(\partial M\) is strictly f-mean convex. We prove that there exists a properly embedded smooth f-minimal surface \(\Sigma \) in M with free boundary \(\partial \Sigma \) on \(\partial M\). If we further assume that the boundary \(\partial M\) is strictly convex, then we prove that \(M^3\) is diffeomorphic to the 3-ball \(B^3\), and a compactness theorem for the space of properly embedded f-minimal surfaces with free boundary in such \((M^3,g,e^{-f}d\mu _M)\), when the topology of these f-minimal surfaces is fixed.     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.