Let T be a tree on n vertices and let be the q-analogue of its Laplacian. For a partition , let the normalized immanant of indexed by λ be denoted as . A string of inequalities among is known when λ varies over hook partitions of n as the size of the first part of λ decreases. In this work, we show a similar sequence of inequalities when λ varies over two row partitions of n as the size of the first part of λ decreases. Our main lemma is an identity involving binomial coefficients and irreducible character values of indexed by two row partitions. Our proof can be interpreted using the combinatorics of Riordan paths and our main lemma admits a nice probabilisitic interpretation involving peaks at odd heights in generalized Dyck paths or equivalently involving special descents in Standard Young Tableaux with two rows. As a corollary, we also get inequalities between and when and are comparable trees in the poset and when and are both two rowed partitions of n, with having a larger first part than .

For a nonnegative potential function q and a given locally finite graph G, we study the combinatorial Schrödinger operator Lq(G)=ΔG+q with Dirichlet boundary condition on a proper finite subset S of the vertex set of G such that the induced subgraph on S is connected. Let Υp={q∈Lp(S):q(x)≥0,∑x∈Sqp(x)≤1}, for 1≤p<∞. We prove the existence and uniqueness of the maximizer of the smallest Dirichlet eigenvalue of Lq(G), whenever the potential function q∈Υp. Furthermore, we also establish the analogue of the Euler–Lagrange equation on graphs.

The combinatorial heat and wave equations on all finite Cayley and coset graphs with discrete time variable was solved by Lal et al. In this paper, the results of the above paper are extended for infinite Cayley and coset graphs, whenever the associated groups are discrete, abelian and finitely generated. Furthermore, we study the solution of the combinatorial heat and wave equations on a k-regular tree whose associated group is a non-abelian free group on k generators, each of order 2. It turns out that in case of Cayley graphs the solutions to combinatorial heat and wave equations are weighted sum of the initial functions over balls of certain radius which are dependent on the discrete time variable.

We supply a combinatorial description of any minor of the adjacency matrix of a graph. This description is then used to give a formula for the determinant and inverse of the adjacency matrix, A(G), of a graph G, whenever A(G) is invertible, where G is formed by replacing the edges of a tree by path bundles.

Let \(\mathbb{K}\) be a finite extension of a characteristic zero field \(\mathbb{F}\). We say that a pair of n × n matrices (A,B) over \(\mathbb{F}\) represents \(\mathbb{K}\) if \(\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle B \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle B \right\rangle }}\), where \(\mathbb{F}\left[ A \right]\) denotes the subalgebra of \(\mathbb{M}_n \left( \mathbb{F} \right)\) containing A and 〈B〉 is an ideal in \(\mathbb{F}\left[ A \right]\), generated by B. In particular, A is said to represent the field \(\mathbb{K}\) if there exists an irreducible polynomial \(q\left( x \right) \in \mathbb{F}\left[ x \right]\) which divides the minimal polynomial of A and \(\mathbb{K} \cong {{\mathbb{F}\left[ A \right]} \mathord{\left/ {\vphantom {{\mathbb{F}\left[ A \right]} {\left\langle {q\left( A \right)} \right\rangle }}} \right. \kern-\nulldelimiterspace} {\left\langle {q\left( A \right)} \right\rangle }}\).

Let G be a connected graph and let L(G) be its Laplacian matrix. We show that given a graph G with a point of articulation u, and a spanning tree T, there is a way to give weights to the edges of G, so that u is the characteristic vertex and the monotonicity property holds on T. A restricted graph is a graph with a restriction that each block can have at most two points of articulation. We supply the structure of a restricted graph G whose algebraic connectivity is extremized among all restricted graphs with the same blocks as those of G. Further results are supplied when each block of G is complete. A path bundle is a graph that consists of internally vertex disjoint paths of the same length with common end vertices. Results pertaining to extremizing the algebraic connectivity of restricted graphs whose blocks are path bundles are supplied. As an application, a comparison of the algebraic connectivities of the sunflower graphs is provided.

A labelling of a graph over a field F, is a mapping of the edge set of the graph into F. A labelling is called magic if for any vertex, the sum of the labels of all the edges incident to it is the same. The class of all such labellings forms a vector space over F and is called the magic space of the graph. For finite graphs, the dimensional structure of the magic space is well known. In this paper, we give the existence of magic labellings and discuss the dimensional structure of the magic space of locally finite graphs. In particular, for a class of locally finite graphs, we give an explicit basis of the magic space.

Building on the work by E. Barletta and S. Dragomir (2002) [3] , this paper solves the initial value problems for the combinatorial heat and wave equations on Cayley and coset graphs.

In this paper we consider the following problem: Over the class of all simple connected graphs of order $n$ with $k$ pendant vertices ($n,k$ being fixed), which graph maximizes (respectively, minimizes) the algebraic connectivity? We also discuss the algebraic connectivity of unicyclic graphs.

An r-labeling of the vertices of a graph $G=(V(G),E(G))$, $f:V(G)\longrightarrow\{1,2,\ldots,r\}$, is said to be distinguishing provided that no nontrivial automorphism of G preserves all of the vertex labels. The distinguishing number of G, denoted by $D(G)$, is the minimum r such that G has a distinguishing r-labeling. The distinguishing chromatic number $\chi_D(G)$ of G is defined similarly, where, in addition, f is assumed to be a proper coloring. In this paper, we determine $D(G)$ and $\chi_D(G)$ of the book graph $B_{m,n}$, and $\chi_D(G)$ of the generalized Petersen graph $P_{n,k}$.

A bidirected tree is a tree in which each edge is replaced by two arcs in either direction. Formulas are obtained for the determinant and the inverse of a bidirected tree, generalizing well-known formulas in the literature.

Let T be a tree, and L(T) be its Laplacian matrix. Let p(T), q(T) be the number of pendant and quasipendant (adjacent to a pendant) vertices of T, respectively. We show that given any tree on n1 vertices and any integer vector y of size n, there is a larger tree T ' with p(T ') = q(T ') and a nonzero integer vector y' such that L(T ')y' =y' and y' restricted to T is y. Let mT(1) denote the multiplicity of the Laplacian eigenvalue 1 of the tree T. Let T be a tree obtained by adding an edge between a vertex of a tree T(1) and a vertex of a tree T(2). We discuss the relationship among m(T1)(1), m(T2)(1), and m(T)(1) and give a condition under which m(T)(1) = m(T1)(1) +m(T2)(1)-1, which generalizes a known result of Grone, R. Merris, R. Sunder, VS. 1990. The Laplacian spectrum of a graph. SIAM J. Matrix Anal. Appl. 11(2) 218-238. We also give a sufficient condition under which m(T)(1) = m(T1)(1)+m(T2)(1). We give a complete characterization of trees that have 1 as the third smallest Laplacian eigenvalue. As an application we characterize trees in which spectral integral variation occurs in one place by adding an edge where the changed eigenvalue is the third smallest eigenvalue.

In this paper we study the class of weakly quasi-threshold graphs that are obtained from a vertex by recursively applying the operations (i) adding a new isolated vertex, (ii) adding a new vertex and making it adjacent to all old vertices, (iii) disjoint union of two old graphs, and (iv) adding a new vertex and making it adjacent to all neighbours of an old vertex. This class contains the class of quasi-threshold graphs. We show that weakly quasi-threshold graphs are precisely the comparability graphs of a forest consisting of rooted trees with each vertex of a tree being replaced by an independent set. We also supply a quadratic time algorithm in the the size of the vertex set for recognizing such a graph. We completely determine the Laplacian spectrum of weakly quasi-threshold graphs. It turns out that weakly quasi-threshold graphs are Laplacian integral. As a corollary we obtain a closed formula for the number of spanning trees in such graphs. A conjecture of Grone and Merris asserts that the spectrum of the Laplacian of any graph is majorized by the conjugate of the degree sequence of the graph. We show that the conjecture holds for cographs.

We prove an upper bound on the covering radius of linear codes over IFq in terms of their generalized Hamming weights. We show that this bound is strengthened if we know that the codes satisfy the chain condition or a partial chain condition. We show that this bound improves all prior bounds. Necessary conditions for equality are also presented. Several applications of our bound are presented. We give tables of improved bounds on the covering radius of many cyclic codes using their generalized Hamming weights. We show that most cyclic codes of length ≤ 39 satisfy the chain condition or partial chain condition up to level 5. We use these results to derive tighter bounds on the covering radius of cyclic codes.

In this article, we consider the following problem: Of all trees on n vertices with diameter d (both fixed) which tree achieves the maximal Laplacian spectral radius? We show that the maximal Laplacian spectral radius is obtained uniquely at , where is a tree obtained by taking a path P on d + 1 vertices and adding n-d-1 pendant vertices to a center point of P.

Artin's braid groups have been recently suggested as a new source for public-key cryptography. In this paper we propose the first undeniable signature schemes using the conjugacy problem and the decomposition problem in the braid groups which are believed to be hard problems.

In an earlier paper the authors studied simplex codes of type α and β over\({\mathbb{Z}}_4\) and obtained some known binary linear and nonlinear codes as Gray images of these codes. In this correspondence, we study weight distributions of simplex codes of type α and β over\({\mathbb{Z}}_{{2^s}}.\) The generalized Gray map is then used to construct binary codes. The linear codes meet the Griesmer bound and a few non-linear codes are obtained that meet the Plotkin/Johnson bound. We also give the weight hierarchies of the first order Reed-Muller codes over\({\mathbb{Z}}_{2^{s}}.\) The above codes are also shown to satisfy the chain condition.

A Coxeter graph is a connected graph each of whose edges is labeled with an integer ⩾ 3 or with ∞. The adjacency matrix of a Coxeter graph G, denoted by A(G) = (aij), is defined to be a square matrix of order |V|, where aij = 2 cos(πp) if the edge (i, j) is labeled with the integer p, and 0 if there is no edge joining vertex i with vertex j. For any positive integer k, we denote by Pk the characteristic polynomial of the adjacency matrix of the path on k vertices. A Coxeter graph G is said to be path-positive if for all positive integers k the matrix Pk(A(G)) is entrywise nonnegative. It is shown that with the exception of a few cases, which are A, B, D, E, F, H, and I, any Coxeter graph is path-positive. The result can be interpreted as a new criterion for the infiniteness of a Coxeter group.

If A is an n×n matrix and q a complex number, then the q-permanent of A is defined as perqA∑σ∈Snql(σ)пni1aiσ(i) where Sn is the symmetric group of degree n and l(σ) denotes the number of inversions of σ [i.e., the number of pairs i,j such that 1⩽i σ(j)]. The function is of interest in that it includes both the determinant and the permanent as special cases. It is known that if A is positive semidefinite and if −1⩽q⩽1, then perqA ⩾ 0. We obtain some results for the q-permanent, including Gram's inequality. It has been conjectured by one of the authors that if A is positive definite and not a diagonal matrix, then perqA is strictly increasing in [−1, 1]. We propose some more conjectures.