We study degree sequences for simplicial posets and polyhedral complexes, generalizing the well-studied graphical degree sequences. Here we extend the more common generalization of vertex-to-facet degree sequences by considering arbitrary face-to-flag degree sequences. In particular, these may be viewed as natural refinements of the flag f-vector of the poset. We investigate properties and relations of these generalized degree sequences, proving linear relations between flag degree sequences in terms of the composition of rank jumps of the flag. As a corollary, we recover an f-vector inequality on simplicial posets first shown by Stanley.
In this paper some properties of generalized tribonacci and generalized Padovan sequence are presented. Also the Euclidean norms of circulant, $r$-circulant, semi-circulant and Hankle matrices with above mentioned sequences are calculated. The upper and lower bounds of spectral norms are also obtained.
We determine the degree sequence of the generalized Sierpinski graph and its general first Zagreb index in terms of the same parameters of the base graph G.
Using sequences of finite length with positive integer elements and the inversion statistic on such sequences, a collection of binomial and multinomial identities are extended to their $q$-analog form via combinatorial proofs. Using the major index statistic on sequences, a connection between integer partitions with kinds and finite differences of the coefficients of generalized Galois numbers is established.
In this note we use the analogy between the Catalan sequence and the Rueppel sequence to derive a variety of conjectures surrounding the Hankel transforms of a number of sequences closely related to the Rueppel sequence. Use is made of the representation of suitable generating functions by Stieltjes continued fractions. We define polynomial sequences by introducing parameters that define generalized Rueppel sequences, and we show that such polynomials have coefficient arrays that are Riordan arrays. Finally we conjecture the form of a product of Hankel transforms arising from the Rueppel seque…
The Tribonacci sequence is a well-known example of third order recurrence sequence, which belongs to a particular class of recursive sequences. In this article, other generalized Tribonacci sequence is introduced and defined by $H_{n+2}=H_{n+1}+H_{n}+H_{n-1}\ \ (n\geq 1)$, where $H_{0}=3$, $H_{1}=0$ and $H_{2}=2$. Also $n$-th power of the generating matrix for this generalized Tribonacci sequence is established and some basic properties of this sequence are obtained by matrix methods. There are many elementary formulae relating the various $H_{n}$, most of which, since the sequence is defined …
We study the possible values of the matching number among all trees with a given degree sequence as well as all bipartite graphs with a given bipartite degree sequence. For tree degree sequences, we obtain closed formulas for the possible values. For bipartite degree sequences, we show the existence of realizations with a restricted structure, which allows to derive an analogue of the Gale-Ryser Theorem characterizing bipartite degree sequences. More precisely, we show that a bipartite degree sequence has a realization with a certain matching number if and only if a cubic number of inequalitie…
In this paper, we characterize the degree sequences excluding the degree sequence of a square in terms of forcibly chordal graphs, and we prove several related results.
In this paper we relate a fundamental parameter of a random graph, its degree sequence, to a simple model of nearly independent binomial random variables. This confirms a conjecture made in 1997. As a result, many interesting functions of the joint distribution of graph degrees, such as the distribution of the median degree, become amenable to estimation. Our result is established by proving an asymptotic formula conjectured in 1990 for the number of graphs with given degree sequence. In particular, this gives an asymptotic formula for the number of $d$-regular graphs for all $d$, as $n\to\inf…
In this paper, we study the abelian complexity $ρ_n^{ab}(\mathbf{t}^{(k)})$ of generalized Thue-Morse sequences $\mathbf{t}^{(k)}$. We obtain the exact value of $ρ_n^{ab}(\mathbf{t}^{(k)})$ for every integer $n\geq k$. Consequently, $ρ_n^{ab}(\mathbf{t}^{(k)})$ is ultimately periodic with the period $k$. Moreover, we show that the abelian complexities of a class of infinite sequences are $k$-automatic.
Many degree sequences can only be realised in graphs that contain a `ds-completable card', defined as a vertex-deleted subgraph in which the erstwhile neighbours of the deleted vertex can be identified from their degrees, if one knows the degree sequence of the original graph. We obtain conditions on the degree sequence, such that graphs whose degree sequence satisfies one of the conditions must contain such a card. The methods allow all such sequences on graphs of order up to 10 to be identified, and some fraction of the sequences for larger graphs. Among other applications, this can be used …
Stanley has conjectured that the h-vector of a matroid complex is a pure O-sequence. We will prove this for cotransversal matroids by using generalized permutohedra. We construct a bijection between lattice points inside a r-dimensional convex polytope and bases of a rank r transversal matroid.
We consider packing tree degree sequences in this paper. We set up a conjecture that any arbitrary number of tree degree sequences without common leaves have edge disjoint tree realizations. This conjecture is known to be true for $2$ and $3$ tree degree sequences. In this paper, we give a proof for $4$ tree degree sequences and a computer aided proof for $5$ tree degree sequences. We also prove that for arbitrary $k$, $k$ tree degree sequences without common leaves and at least $2k-4$ vertices which are not leaves in any of the trees always have edge disjoint tree realizations. The main ingre…
In this paper, we give quadratic approximation of generalized Tribonacci sequence $\{V_{n}\}_{n\geq0}$ defined by Eq. (\ref{eq:7}) and use this result to give the matrix form of the $n$-th power of a companion matrix of $\{V_{n}\}_{n\geq0}$. Then we re-prove the cubic identity or Cassini-type formula for $\{V_{n}\}_{n\geq0}$ and the Binet's formula of the generalized Tribonacci quaternions.
We consider special multiclass spectral, discrepancy, degree, and codegree properties of expanding graph sequences. As we can prove equivalences and implications between them and the definition of the generalized quasirandomness of Lovász--Sós (2008), they can be regarded as generalized quasirandom properties akin to the equivalent quasirandom properties of the seminal Chung--Graham--Wilson paper (1989) in the one-class scenario. Since these properties are valid for certain deterministic graph sequences, irrespective of stochastic models, the partial implications also justify for law-dimension…
Holonomic Sequences Arising from Laplace Transforms of Generalized Second Order Sequences
The Citadel（Citadel）；Escuela de Matemáticas y Estadística, Universidad Pedagógica y Tecnológica de Colombia（哥伦比亚教育与技术大学数学与统计学院）；Departamento de Matemáticas, Universidad Nacional de Colombia（哥伦比亚国立大学数学系）
There are typically several nonisomorphic graphs having a given degree sequence, and for any two degree sequence terms it is often possible to find a realization in which the corresponding vertices are adjacent and one in which they are not. We provide necessary and sufficient conditions for two vertices to be adjacent (or nonadjacent) in every realization of the degree sequence. These conditions generalize degree sequence and structural characterizations of the threshold graphs, in which every adjacency relationship is forcibly determined by the degree sequence. We further show that degree se…
The degree sequence of a graph is a numerical method to characterize the properties of graphs. Generalized forms of degree sequences exist for complete graphs and complete graphs. Nikolopolus et al. characterized the number of spanning trees from edge deletions and edge additions. Instead of investigating the number of spanning trees of graphs that arise from edge additions and deletions, we sought to characterize degree sequences of such graphs. We conjecture a characterization for the degree sequence of the addition and edge deletion for many families of graphs including threshold graphs and…
Bidirected graphs generalize directed and undirected graphs in that edges are oriented locally at every node. The natural notion of the degree of a node that takes into account (local) orientations is that of net-degree. In this paper, we extend the following four topics from (un)directed graphs to bidirected graphs: - Erdős-Gallai-type results: characterization of net-degree sequences, - Havel-Hakimi-type results: complete sets of degree-preserving operations, - Extremal degree sequences: characterization of uniquely realizable sequences, and - Enumerative aspects: counting formulas for net-d…
The switch chain is a well-studied Markov chain which can be used to sample approximately uniformly from the set $Ω(\boldsymbol{d})$ of all graphs with a given degree sequence $\boldsymbol{d}$. Polynomial mixing time (rapid mixing) has been established for the switch chain under various conditions on the degree sequences. Amanatidis and Kleer introduced the notion of strongly stable families of degree sequences, and proved that the switch chain is rapidly mixing for any degree sequence from a strongly stable family. Using a different approach, Erdős et al. recently extended this result to the …
Necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a k-connected simple graph are detailed. Conditions are also given under which such a sequence is necessarily k-connected.
In this paper, we present several upper bounds for the adjacency and signless Laplacian spectral radii of uniform hypergraphs in terms of degree sequences.
The degree partition of a simple graph is its degree sequence rearranged in weakly decreasing order. The polytope of degree partitions (respectively, degree sequences) is the convex hull of all degree partitions (respectively, degree sequences) of a fixed length. We think of the degree sequence polytope as the symmetrization of the degree partition polytope and the degree partition polytope as the asymmetric part of the degree sequence polytope. The degree sequence polytope is a well studied object with formulas (generating functions) known for its face numbers, volume, and number of lattice p…
This note gives necessary and sufficient conditions for a sequence of non-negative integers to be the degree sequence of a connected simple graph. This result is implicit in a paper of Hakimi. A new alternative characterisation of these necessary and sufficient conditions is also given.
We define sequence patterns of length $n$ and level $\ell$ to be equivalence classes of sequences that have $n$ elements from the set of $\ell$ integer symbols $\{1,2,\ldots,\ell\}$ with no restriction on repetition, where the equivalence relation is induced by symbol relabeling without swapping positions of symbols. We define a distance for a set of $k$ sequence patterns of length $n$ and level $\ell$ by generalizing the Hamming distance between sequences. We compute the maximal distance for $k$ sequence patterns of length $n$ and level $\ell$ and demonstrate how to calculate the exact distan…
A fundamental result of Kühn and Osthus [The minimum degree threshold for perfect graph packings, Combinatorica, 2009] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect H-tiling. We prove a degree sequence version of this result which allows for a significant number of vertices to have lower degree.
The Erdős--Gallai criteria for recognizing degree sequences of simple graphs involve a system of inequalities. Given a fixed degree sequence, we consider the list of differences of the two sides of these inequalities. These differences have appeared in varying contexts, including characterizations of the split and threshold graphs, and we survey their uses here. Then, enlarging upon properties of these graph families, we show that both the last term and the maximum term of the principal Erdős--Gallai differences of a degree sequence are preserved under graph complementation and are monotonic u…
We introduce fractional realizations of a graph degree sequence and a closely associated convex polytope. Simple graph realizations correspond to a subset of the vertices of this polytope. We describe properties of the polytope vertices and characterize degree sequences for which each polytope vertex corresponds to a simple graph realization. These include the degree sequences of pseudo-split graphs, and we characterize their realizations both in terms of forbidden subgraphs and graph structure.
We find an asymptotic enumeration formula for the number of simple $r$-uniform hypergraphs with a given degree sequence, when the number of edges is sufficiently large. The formula is given in terms of the solution of a system of equations. We give sufficient conditions on the degree sequence which guarantee existence of a solution to this system. Furthermore, we solve the system and give an explicit asymptotic formula when the degree sequence is close to regular. This allows us to establish several properties of the degree sequence of a random $r$-uniform hypergraph with a given number of edg…
The skew of a binary string is the difference between the number of zeroes and the number of ones, while the length of the string is the sum of these two numbers. We consider certain suffixes of the lexicographically-least de Bruijn sequence at natural breakpoints of the binary string. We show that the skew and length of these suffixes are enumerated by sequences generalizing the Fibonacci and Lucas numbers, respectively.
In this paper, we asymptotically enumerate graphs with a given degree sequence d=(d_1,...,d_n) satisfying restrictions designed to permit heavy-tailed sequences in the sparse case (i.e. where the average degree is rather small). Our general result requires upper bounds on functions of M_k= \sum_{i=1}^n [d_i]_k for a few small integers k\ge 1. Note that M_1 is simply the total degree of the graphs. As special cases, we asymptotically enumerate graphs with (i) degree sequences satisfying M_2=o(M_1^{ 9/8}); (ii) degree sequences following a power law with parameter gamma>5/2; (iii) power-law degr…
The Wiener index of a connected graph is the sum of topological distances between all pairs of vertices. Since Wang gave a mistake result on the maximum Wiener index for given tree degree sequence, in this paper, we investigate the maximum Wiener index of trees with given degree sequences and extremal trees which attain the maximum value.
We consider special cases of the two tree degree sequences problem. We show that if two tree degree sequences do not have common leaves then they always have edge-disjoint caterpillar realizations. By using a probabilistic method, we prove that two tree degree sequences always have edge-disjoint realizations if each vertex is a leaf in at least one of the trees. This theorem can be extended to more trees: we show that the edge packing problem is in P for an arbitrary number of tree sequences with the property that each vertex is a non-leaf in at most one of the trees. We also consider the foll…
Given the degree sequence $d$ of a graph, the realization graph of $d$ is the graph having as its vertices the labeled realizations of $d$, with two vertices adjacent if one realization may be obtained from the other via an edge-switching operation. We describe a connection between Cartesian products in realization graphs and the canonical decomposition of degree sequences described by R.I. Tyshkevich and others. As applications, we characterize the degree sequences whose realization graphs are triangle-free graphs or hypercubes.
It is well known that the set of possible degree sequences for a graph on $n$ vertices is the intersection of a lattice and a convex polytope. We show that the set of possible degree sequences for a $k$-uniform hypergraph on $n$ vertices is not the intersection of a lattice and a convex polytope for $k \geq 3$ and $n \geq k+13$. We also show an analogous nonconvexity result for the set of degree sequences of $k$-partite $k$-uniform hypergraphs and the generalized notion of $λ$-balanced $k$-uniform hypergraphs.
In this paper, we investigate the structures of an extremal tree which has the minimal number of subtrees in the set of all trees with the given degree sequence of a tree. In particular, the extremal trees must be caterpillar and but in general not unique. Moreover, all extremal trees with a given degree sequence $π=(d_1, ..., d_5, 1,..., 1)$ have been characterized.
Since 1997 a considerable effort has been spent on the study of the swap (switch) Markov chains on graphic degree sequences. Several results were proved on rapidly mixing Markov chains on regular simple, on regular directed, on half-regular directed and on half-regular bipartite degree sequences. In this paper, the main result is the following: Let $U$ and $V$ be disjoint finite sets, and let $0 < c_1 \le c_2 < |U|$ and $0 < d_1 \le d_2 < |V|$ be integers. Furthermore, assume that the bipartite degree sequence on $U \cup V$ satisfies $c_1 \le d(v) \le c_2\ : \ \forall v\in V$ and $d_1 \le d(u)…
Necessary and sufficient conditions for a sequence of positive integers to be the degree sequence of a 3-connected simple graph are detailed. Conditions are also given under which such a sequence is necessarily 3-connected i.e. the sequence can only be realised as a 3-connected graph. Finally, a matrix is introduced whose non-empty entries partition the set of 3-connected graphs.
We consider $m$-th order linear recurrences that can be thought of as generalizations of the Lucas sequence. We exploit some interplay with matrices that again can be considered generalizations of the Fibonacci matrix. We introduce the definition of reflected sequence and inverted sequence and we establish some relationship between the coefficients of the Cayley-Hamilton equation for these matrices and the introduced sequences.
Horadam introduced a new generalized sequence of numbers, describing its key features and the special sub-sequences that are obtained depending on the choices of initial parameters. This sequence and its sub-sequences are known as the Horadam, generalized Fibonacci, and generalized Lucas numbers, respectively. In the present study, we propose another new sequence, which satisfies a second-order recurrence relation, in addition to Horadam's definition. Further, we prove the Binet's formula, some famous identities, and summation formulas for this new sequence. In particular, we demonstrate the i…
S.B. Rao conjectured in 1971 that graphic degree sequences are well quasi ordered by a relation defined in terms of the induced subgraph relation. In 2008, M. Chudnovsky and P. Seymour proved this long standing Rao's Conjecture by giving structure theorems for graphic degree sequences. In this paper, we prove and use a variant of Dickson's Lemma from commutative semigroup theory to give a short proof of the bounded degree case of Rao's Conjecture that is independent of the Chudnovsky-Seymour structure theory. In fact, we affirmatively answer two questions of N. Robertson, the first of which im…
We give sufficient conditions under which a random graph with a specified degree sequence is symmetric or asymmetric. In the case of bounded degree sequences, our characterisation captures the phase transition of the symmetry of the random graphs. This phase transition coincides with that of the graph connectivity. However, when the maximum degree is a growing function as the number of vertices tends to infinity, our results suggest that these two thresholds do not coincide any more
We explore the notion of degree of asymmetry for integer sequences and related combinatorial objects. The degree of asymmetry is a new combinatorial statistic that measures how far an object is from being symmetric. We define this notion for compositions, words, matchings, binary trees and permutations, we find generating functions enumerating these objects with respect to their degree of asymmetry, and we describe the limiting distribution of this statistic in each case.
For a symmetric bivariable function $f(x,y)$, let the {\it connectivity function} of a connected graph $G$ be $M_f(G)=\sum_{uv\in E(G)}f(d(u),d(v))$, where $d(u)$ is the degree of vertex $u$. In this paper, we prove that for an escalating (de-escalating) function $f(x,y)$, there exists a BFS-graph with the maximum (minimum) connectivity function $M_f(G)$ among all graphs with a $c-$cyclic degree sequence $π=(d_1,d_2, \ldots, d_n)$ and $d_n=1$, and obtain the majorization theorem for connectivity function for unicyclic and bicyclic degree sequences. Moreover, some applications of graph invarian…
The deck of a graph $G$ is the multiset of cards $\{G-v:v\in V(G)\}$. Myrvold (1992) showed that the degree sequence of a graph on $n\geq7$ vertices can be reconstructed from any deck missing one card. We prove that the degree sequence of a graph with average degree $d$ can reconstructed from any deck missing $O(n/d^3)$ cards. In particular, in the case of graphs that can be embedded on a fixed surface (e.g. planar graphs), the degree sequence can be reconstructed even when a linear number of the cards are missing.
This paper investigates some properties of the number of subtrees of a tree with given degree sequence. These results are used to characterize trees with the given degree sequence that have the largest number of subtrees, which generalizes the recent results of Kirk and Wang. These trees coincide with those which were proven by Wang and independently Zhang et al. to minimize the Wiener index. We also provide a partial ordering of the extremal trees with different degree sequences, some extremal results follow as corollaries.
We study the generalized Hankel transform of the family of sequences satisfying the recurrence relation $a_{n+1} = \bigl(α+ \fracβ{n+γ}\bigr) a_n$. We apply the obtained formula to several particular important sequences. Incidentally, we find a connection between some well known formulas that had previously arisen in literature in dissimilar settings. Additionally, given a non-zero sequence satisfying the above recurrence, we evaluate the Hankel transform of the sequence of its reciprocals.
We give a simplified presentation of some results about recurrences of certain sequences of binomial sums in terms of (generalized) Fibonacci and Lucas polynomials.
We show that if the degree sequence of a graph $G$ is close in $\ell_1$-distance to a given realizable degree sequence $(d_1,\dots,d_n)$, then $G$ is close in edit distance to a graph with degree sequence $(d_1,\dots,d_n)$. We then use this result to prove that every graph property defined in terms of the degree sequence is testable in the dense graph model with query complexity independent of $n$.
We consider sequences of the type $A_n=6A_{n-1}-A_{n-2}, A_0=r, A_1=s$ ($r$ and $s$ integers) and show that all sequences that solve particular cases of the Pell generalized equation are expressible as a constant times one of four particular sequences of the same type.
Let $F(z)$ be a $k$-regular series in $\mathbb{Z}[[z]]$ and $b$ be an integer with $b\ge2$. Bell, Bugeaud and Coons [BelBC] proved that $F(\frac{1}{b})$ is either rational or transcendental. In [Mi], we introduce a generalized $k$-regular sequence as a unification of several kinds of important sequences including $k$-regular, $k$-additive and $k$-multiplicative sequences. In this paper, we give a generalization of the result of Bell, Bugeaud and Coons for certain generalized $k$-regular series. Especially, we show that the values of irrational generating functions of certain sum of $k$-additiv…
We establish a generalized Ihara zeta function formula for simple graphs with bounded degree. This is a generalization of the formula obtained by G. Chinta, J. Jorgenson and A. Karlsson from a vertex-transitive graph.
