Sequence D1 Research Articles

Maximizing the degree powers of graphs with fixed size

For a graph G with degree sequence (d1,d2,…,dn) and a positive integer p, denote the degree power of G by ep(G)=∑i=1ndip. In this paper, we concentrate on the extremal problems of degree powers in graphs with fixed size. By the majorization theory, we firstly characterize the graphs with the maximum ep(G) among all connected graphs in G(n,m) where m≤2n−3. The second main result is about the graphs with forbidden cycles. Concretely, we determine the maximum ep(G) among all Cℓ-free graphs in G(m) and characterize the corresponding extremal graphs. An analogous result is obtained in non-bipartite graphs. Furthermore, we respectively give the sharp upper bounds on ep(G) in G(m) with respect to various graph parameters, such as clique number, chromatic number, girth, and circumference.

Structural properties of Toeplitz graphs

In this paper, we study structural properties of Toeplitz graphs. We characterize Kq-free Toeplitz graphs for an integer q≥3 and give equivalent conditions for a Toeplitz graph Gn〈t1,t2,…,tk〉 with t1<⋯<tk and n≥tk−1+tk being chordal and equivalent conditions for a Toeplitz graph Gn〈t1,t2〉 being perfect. Then we compute the edge clique cover number and the vertex clique cover number of a chordal Toeplitz graph. Finally, we characterize the degree sequence (d1,d2,…,dn) of a Toeplitz graph with n vertices and show that a Toeplitz graph is a regular graph if and only if it is a circulant graph.

Degree powers in Ks,t-minor free graphs

Let G be a simple graph with the degree sequence (d1,d2,…,dn). Given a positive integer p, denote the degree power of G by ep(G)=∑i=1ndip. In this paper, we study the following problem: what is the maximum value of ep(G) where G is a Ks,t-minor free graph of order n? By using majorization, we solve the problem for K2,3-minor free and K3,3-minor free graphs and characterize the extremal graphs. The results generalize the results of outerplanar [28] and planar graphs [18,28]. Ding, Johnson and Seymour [7] proved the maximum value of e1(G) when G is K1,t-minor free. We generalize the result by showing the maximum value of ep(G) for p≥2 among all K1,t-minor free graphs and characterizing the extremal graph. Further, we determine the maximum degree power for bipartite planar graphs and the minimum degree power for non-planar graphs, and the corresponding extremal graphs are also determined. In the end, we conclude this paper with an open problem.

Maximum bisections of graphs without short even cycles

A bisection of a graph is a bipartition of its vertex set in which the number of vertices in the two parts differ by at most 1, and its size is the number of edges which go across the two parts. In this paper, motivated by several known results of Alon, Bollobás, Krivelevich and Sudakov about Max-Cut, we study maximum bisections of graphs without short even cycles. Let G be a graph on m edges without cycles of length 4 and 6. We first extend a well-known result of Shearer on maximum cuts to bisections and show that if G has a perfect matching and degree sequence d1,…,dn, then G admits a bisection of size at least m/2+Ω(∑i=1ndi). This is tight for certain polarity graphs. Together with a technique of Nikiforov, we prove that if G also contains no cycle of length 2k≥6 then G either has a large bisection or is nearly bipartite. As a corollary, if G has a matching of size Θ(n), then G admits a bisection of size at least m/2+Ω(m(2k+1)/(2k+2)) and that this is tight for 2k∈{6,10}; if G has a matching of size o(n), then the bound remains valid for G with minimum degree at least 2.

On graph irregularity indices with particular regard to degree deviation

Let G = (V,E), V = {v1, v2,..., vn}, be a simple connected graph of order n and size m, with vertex degree sequence d1 ? d2 ? ...? dn. A graph G is said to be regular if d1 = d2 =... = dn. Otherwise it is irregular. In many applications and problems it is important to know how irregular a given graph is. A quantity called degree deviation S(G) = ?ni=1 |di-2m/n| can be used as an irregularity measure. Some new lower bounds for S(G) are obtained. A simple formula for computing S(G) for connected bidegreed graphs is derived also. Besides, two novel irregularity measures are introduced.

Some mathematical properties of the geometric-arithmetic index/coindex of graphs

Let G = (V,E), V = {1,2,...,n}, be a simple connected graph of order n, size m with vertex degree sequence d1 ? d2 ? ... ? dn &gt; 0, di = d(vi). The geometric-arithmetic topological index of G is defined as GA(G) = ? i~j 2? didj/di+dj, whereas the geometric-arithmetic coindex as GA?(G) = ? i~/j 2 ? didj/di+dj . New lower bounds for GA(G) and GA(G) in terms of some graph parameters and other invariants are obtained.

Subgraph counts for dense random graphs with specified degrees

AbstractWe prove two estimates for the expectation of the exponential of a complex function of a random permutation or subset. Using this theory, we find asymptotic expressions for the expected number of copies and induced copies of a given graph in a uniformly random graph with degree sequence(d1, …, dn) as n→ ∞. We also determine the expected number of spanning trees in this model. The range of degrees covered includes dj= λn + O(n1/2+ε) for some λ bounded away from 0 and 1.

Primitive divisors of elliptic divisibility sequences over function fields with constant j-invariant

We prove an optimal Zsigmondy bound for elliptic divisibility sequences over function fields in case the j-invariant of the elliptic curve is constant.In more detail, given an elliptic curve E with a point P of infinite order over a global field, the sequence D1, D2,… of denominators of multiples P, 2P,… of P is a strong divisibility sequence in the sense that gcd⁡(Dm,Dn)=Dgcd⁡(m,n). This is the genus-one analogue of the genus-zero Fibonacci, Lucas and Lehmer sequences.A number N is called a Zsigmondy bound of the sequence if each term Dn with n>N presents a new prime factor. The optimal uniform Zsigmondy bound for the genus-zero sequences over Q is 30 by Bilu-Hanrot-Voutier [2], but finding such a bound remains an open problem in genus one, both over Q and over function fields.We prove that the optimal Zsigmondy bound for ordinary elliptic divisibility sequences over function fields is 2 if the j-invariant is constant. In the supersingular case, we give a complete classification of which terms can and cannot have a new prime factor.

Remarks on general zeroth-order Randić and general sum-connectivity indices

Let G = (V,E ), V = [v \,v2, . . . , vn}, be a simple connected graph with n vertices, m edges and vertex degree sequence d1 > d2 > ■■■ > dn > 0, di = d(vi ). General zeroth-order Randic index of G is defined as °Ra (G) = Ση=ι > and general sum-connectivity index as Xa(G) = (di + d j)α, where a is an arbitrary real number. In this paper we establish a relationship between 0Rα+β (G), ^ α -β ^ ) and °Ra (G), as well as χ α+β (G), χ α -β ^ ) and Xa (G), where α and β are arbitrary real numbers. By the appropriate choice of parameters α and β, a number of new/old inequalities that reveal relationships between various vertex and edge degree-based topological indices are obtained.

Degree powers in graphs with a forbidden forest

Given a positive integer p and a graph G with degree sequence d1,…,dn, we define ep(G)=∑i=1ndip. Caro and Yuster introduced a Turán-type problem for ep(G): Given a positive integer p and a graph H, determine the function exp(n,H), which is the maximum value of ep(G) taken over all graphs G on n vertices that do not contain H as a subgraph. Clearly, ex1(n,H)=2ex(n,H), where ex(n,H) denotes the classical Turán number. Caro and Yuster determined the function exp(n,Pℓ) for sufficiently large n, where p≥2 and Pℓ denotes the path on ℓ vertices. In this paper, we generalise this result and determine exp(n,F) for sufficiently large n, where p≥2 and F is a linear forest. We also determine exp(n,S), where S is a star forest; and exp(n,B), where B is a broom graph with diameter at most six.

Some comments on the Slater number

Let G be a graph with degree sequence d1≥⋯≥dn. Slater proposed sℓ(G)=min{s:(d1+1)+⋯+(ds+1)≥n} as a lower bound on the domination number γ(G) of G. We show that deciding the equality of γ(G) and sℓ(G) for a given graph G is NP-complete but that one can decide efficiently whether γ(G)>sℓ(G) or γ(G)≤lnn(G)sℓ(G)+1sℓ(G). For real numbers α and β with α≥max{0,β}, let G(α,β) be the class of non-null graphs G such that every non-null subgraph H of G has at most αn(H)−β many edges. Generalizing a result of Desormeaux, Haynes, and Henning, we show that γ(G)≤(2α+1)sℓ(G)−2β for every graph G in G(α,β) with α≤32. Furthermore, we show that γ(G)∕sℓ(G) is bounded for graphs G in G(α,β) if and only if α<2. For an outerplanar graph G with sℓ(G)≥2, we show γ(G)≤6sℓ(G)−6. In analogy to sℓ(G), we propose sℓt(G)=min{s:d1+⋯+ds≥n} as a lower bound on the total domination number. Strengthening results due to Raczek as well as Chellali and Haynes, we show that sℓt(T)≥n+2−n12 for every tree T of order n at least 2 with n1 endvertices.

Remark on lower bound for forgotten topological index

Let G be a simple connected graph with n vertices and m edges with vertex degree sequence d1 ≥ d2 ≥ . . . ≥ dn > 0. Denote by F = Σni =1 d3i forgotten topological index of graph G. In this paper we give some lower bounds for invariant F. Also, obtained bounds are compared with some known bounds from the literature.

Fractional spanning tree packing, forest covering and eigenvalues

We investigate the relationship between the eigenvalues of a graph G and fractional spanning tree packing and coverings of G. Let ω(G) denote the number of components of a graph G. The strength η(G) and the fractional arboricity γ(G) are defined by η(G)=min|X|ω(G−X)−ω(G),andγ(G)=max|E(H)||V(H)|−1, where the optima are taken over all edge subsets X whenever the denominator is non-zero. The well known spanning tree packing theorem by Nash-Williams and Tutte indicates that a graph G has k edge-disjoint spanning tree if and only if η(G)≥k; and Nash-Williams proved that a graph G can be covered by at most k forests if and only if γ(G)≤k. Let λi(G) (μi(G), qi(G), respectively) denote the ith largest adjacency (Laplacian, signless Laplacian, respectively) eigenvalue of G. In this paper, we prove the following.(1) Let G be a graph with δ≥2s/t. Then η(G)≥s/t if μn−1(G)>2s−1t(δ+1), or if λ2(G)<δ−2s−1t(δ+1), or if q2(G)<2δ−2s−1t(δ+1).(2) Suppose that G is a graph with nonincreasing degree sequence d1,d2,…,dn and n≥⌊2st⌋+1. Let β=2st−1⌊2st⌋+1∑i=1⌊2st⌋+1di. Then γ(G)≤s/t, if β≥1, or if 0<β<1, n>⌊2s/t⌋+1+2s−2tβ and μn−1(G)>n(2s/t−2/t−β(⌊2s/t⌋+1))(⌊2s/t⌋+1)(n−⌊2s/t⌋−1). Our result proves a stronger version of a conjecture by Cioabă and Wong on the relationship between eigenvalues and spanning tree packing, and sharpens former results in this area.

On some irregularity measures of graphs

Let Γ(G) be a set of all simple graphs of order n and size m, without isolated vertices, with vertex degree sequence d1 ≥ d2 ≥ ... ≥ dn > 0. A graph G is regular if and only if d1 = d2 = ... = dn . Each mapping Irr: Γ(G) ›→ [0, +∞) with the property Irr(G) = 0 if and only if G is regular, is referred to as irregularity measure of graph. In this paper we introduce some new irregularity measures and inequalities that establish relations between them.

On a degree sequence analogue of Pósa's conjecture

A famous conjecture of Pósa from 1962 asserts that every graph on n vertices and with minimum degree at least 2n/3 contains the square of a Hamilton cycle. The conjecture was proven for large graphs in 1996 by Komlós, Sárközy and Szemerédi [Komlós, J., G.N. Sárközy and E. Szemerédi, On the square of a Hamiltonian cycle in dense graphs, Random Structures and Algorithms 9 (1996), 193–211]. We prove a degree sequence version of Pósa's conjecture: Given any η>0, every graph G of sufficiently large order n contains the square of a Hamilton cycle if its degree sequence d1≤⋯≤dn satisfies di≥(1/3+η)n+i for all i≤n/3. The degree sequence condition here is asymptotically best possible. Our approach uses a hybrid of the Regularity-Blow-up method and the Connecting-Absorbing method.

The determination on weight hierarchies of q-ary linear codes of dimension 5 in class IV

The weight hierarchy of a linear [n; k; q] code C over GF(q) is the sequence (d1, d2, ···, dk) where dr is the smallest support of any r-dimensional subcode of C. “Determining all possible weight hierarchies of general linear codes” is a basic theoretical issue and has important scientific significance in communication system. However, it is impossible for q-ary linear codes of dimension k when q and k are slightly larger, then a reasonable formulation of the problem is modified as: “Determine almost all weight hierarchies of general q-ary linear codes of dimension k”. In this paper, based on the finite projective geometry method, the authors study q-ary linear codes of dimension 5 in class IV, and find new necessary conditions of their weight hierarchies, and classify their weight hierarchies into 6 subclasses. The authors also develop and improve the method of the subspace set, thus determine almost all weight hierarchies of 5-dimensional linear codes in class IV. It opens the way to determine the weight hierarchies of the rest two of 5-dimensional codes (classes III and VI), and break through the difficulties. Furthermore, the new necessary conditions show that original necessary conditions of the weight hierarchies of k-dimensional codes were not enough (not most tight nor best), so, it is important to excogitate further new necessary conditions for attacking and solving the k-dimensional problem.

Irregularity measures of graph

Let G = (V,E), V = {1,2 …, n}, be a simple graph without isolated vertices, with vertex degree sequence d1 ≥ d2 ≥ … ≥ dn > 0, di = d(i). A graph G is regular if and only if d1 = d2 = … = dn. A graph invariant I(G) is measure of irregularity of graph G with the property I(G)=0 if and only if G is regular, and I(G)>0 otherwise. In this paper we introduce some new irregularity measures.

The top ten values of harmonic index in chemical trees

Let G be an n-vertex graph with degree sequence d1, d2, …, dn. The harmonic index H(G) is defined as n/I(G)= Σ n i= 1(l/di)., In this paper the top ten values of harmonic index in the set of all chemical trees of order n are determined.

A class of the hamming weight hierarchy of linear codes with dimension 5

The weight hierarchy of a [n; kI q] linear code C over Fq is the sequence(d1,… dr,… dk), where dr is the smallest support weight of an r-dimensional subcode of C. In this paper, by using the finite projective geometry method, we research a class of weight hierarchy of linear codes with dimension 5. We first find some new preconditions of this class. Then we divide its weight hierarchies into six subclasses, and research one subclass to determine nearly all the weight hierarchies of this subclass of weight hierarchies of linear codes with dimension 5.

On the ascending subgraph decomposition problem for bipartite graphs

The Ascending Subgraph Decomposition (ASD) Conjecture asserts that every graph G with (n+12) edges admits an edge decomposition G=H1⊕⋯⊕Hn such that Hi has i edges and is isomorphic to a subgraph of Hi+1, i=1,…,n−1. We show that every bipartite graph G with (n+12) edges such that the degree sequence d1,…,dk of one of the stable sets satisfies di≥n−i+2, 1≤i<k, admits an ascending subgraph decomposition with star forests. We also give a necessary condition on the degree sequence which is not far from the above sufficient one.