The Cyclomatic Number of a Graph and its Independence Polynomial at −1

If by s _k is denoted the number of independent sets of cardinality k in a graph G, then ${I(G;x)=s_{0}+s_{1}x+\cdots+s_{\alpha}x^{\alpha}}$ is the independence polynomial of G (Gutman and Harary in Utilitas Mathematica 24:97–106, 1983), where α = α(G) is the size of a maximum independent set. The inequality |I (G; ?1)| ≤ 2 ^ν(G), where ν(G) is the cyclomatic number of G, is due to (Engström in Eur. J. Comb. 30:429–438, 2009) and (Levit and Mandrescu in Discret. Math. 311:1204–1206, 2011). For ν(G) ≤ 1 it means that ${I(G;-1)\in\{-2,-1,0,1,2\}.}$ In this paper we prove that if G is a unicyclic well-covered graph different from C ₃, then ${I(G;-1)\in\{-1,0,1\},}$ while if G is a connected well-covered graph of girth ≥ 6, non-isomorphic to C ₇ or K ₂ (e.g., every well-covered tree ≠ K ₂), then I (G; ?1) = 0. Further, we demonstrate that the bounds {?2 ^ν(G), 2 ^ν(G)} are sharp for I (G; ?1), and investigate other values of I (G; ?1) belonging to the interval [?2 ^ν(G), 2 ^ν(G)].     

On the roots of independence polynomials of almost all very well-covered graphs

If s_k denotes the number of stable sets of cardinality k in graph G, and α(G) is the size of a maximum stable set, then is the independence polynomial of G [I. Gutman, F. Harary, Generalizations of the matching polynomial, Utilitas Math. 24 (1983) 97-106]. A graph G is very well-covered [O. Favaron, Very well-covered graphs, Discrete Math. 42 (1982) 177-187] if it has no isolated vertices, its order equals 2α(G) and it is well-covered, i.e., all its maximal independent sets are of the same size [M.D. Plummer, Some covering concepts in graphs, J. Combin. Theory 8 (1970) 91-98]. For instance, appending a single pendant edge to each vertex of G yields a very well-covered graph, which we denote by G^*. Under certain conditions, any well-covered graph equals G^* for some G [A. Finbow, B. Hartnell, R.J. Nowakowski, A characterization of well-covered graphs of girth 5 or greater, J. Combin. Theory Ser B 57 (1993) 44-68].The root of the smallest modulus of the independence polynomial of any graph is real [J.I. Brown, K. Dilcher, R.J. Nowakowski, Roots of independence polynomials of well-covered graphs, J. Algebraic Combin. 11 (2000) 197-210]. The location of the roots of the independence polynomial in the complex plane, and the multiplicity of the root of the smallest modulus are investigated in a number of articles.In this paper we establish formulae connecting the coefficients of I(G;x) and I(G^*;x), which allow us to show that the number of roots of I(G;x) is equal to the number of roots of I(G^*;x) different from -1, which appears as a root of multiplicity α(G^*)-α(G) for I(G^*;x). We also prove that the real roots of I(G^*;x) are in [-1,-1/2α(G^*)), while for a general graph of order n we show that its roots lie in |z|>1/(2n-1).Hoede and Li [Clique polynomials and independent set polynomials of graphs, Discrete Math. 125 (1994) 219-228] posed the problem of finding graphs that can be uniquely defined by their clique polynomials (clique-unique graphs). Stevanovic [Clique polynomials of threshold graphs, Univ. Beograd Publ. Elektrotehn. Fac., Ser. Mat. 8 (1997) 84-87] proved that threshold graphs are clique-unique. Here, we demonstrate that the independence polynomial distinguishes well-covered spiders among well-covered trees.     

Critical Independent Sets and König–Egerváry Graphs

A set S of vertices is independent or stable in a graph G, and we write S ∈ Ind (G), if no two vertices from S are adjacent, and α(G) is the cardinality of an independent set of maximum size, while core(G) denotes the intersection of all maximum independent sets. G is called a König–Egerváry graph if its order equals α(G) + μ(G), where μ(G) denotes the size of a maximum matching. The number def (G) = | V(G) | ?2μ(G) is the deficiency of G. The number \({d(G)=\max\{\left\vert S\right\vert -\left\vert N(S)\right\vert :S\in\mathrm{Ind}(G)\}}\) is the critical difference of G. An independent set A is critical if \({\left\vert A\right\vert -\left\vert N(A)\right\vert =d(G)}\) , where N(S) is the neighborhood of S, and α _c (G) denotes the maximum size of a critical independent set. Larson (Eur J Comb 32:294–300, 2011) demonstrated that G is a König–Egerváry graph if and only if there exists a maximum independent set that is also critical, i.e., α _c (G) = α(G). In this paper we prove that: (i) \({d(G)=\left \vert \mathrm{core}(G) \right \vert -\left \vert N (\mathrm{core}(G))\right\vert =\alpha(G)-\mu(G)=def \left(G\right)}\) holds for every König–Egerváry graph G; (ii) G is König–Egerváry graph if and only if each maximum independent set of G is critical.     

Super Edge-connectivity and Zeroth-order General Randić Index for −1 ≤ <Emphasis Type="Italic">α</Emphasis> &lt; 0

Let G be a connected graph with order n, minimum degree δ = δ(G) and edge-connectivity λ = λ(G). A graph G is maximally edge-connected if λ = δ, and super edge-connected if every minimum edgecut consists of edges incident with a vertex of minimum degree. Define the zeroth-order general Randi? index \(R_\alpha ^0\left( G \right) = \sum\limits_{x \in V\left( G \right)} {d_G^\alpha \left( x \right)} \), where d_G(x) denotes the degree of the vertex x. In this paper, we present two sufficient conditions for graphs and triangle-free graphs to be super edge-connected in terms of the zeroth-order general Randi? index for ?1 ≤ α < 0, respectively.     

On a classical theorem on the diameter and minimum degree of a graph

In this work, we obtain good upper bounds for the diameter of any graph in terms of its minimum degree and its order, improving a classical theorem due to Erd¨os, Pach, Pollack and Tuza.We use these bounds in order to study hyperbolic graphs(in the Gromov sense). To compute the hyperbolicity constant is an almost intractable problem, thus it is natural to try to bound it in terms of some parameters of the graph. Let H(n, δ_0) be the set of graphs G with n vertices and minimum degree δ_0, and J(n, Δ) be the set of graphs G with n vertices and maximum degree Δ. We study the four following extremal problems on graphs: a(n, δ_0) = min{δ(G) | G ∈ H(n, δ_0)}, b(n, δ_0) = max{δ(G) |G ∈ H(n, δ_0)}, α(n, Δ) = min{δ(G) | G ∈ J(n, Δ)} and β(n, Δ) = max{δ(G) | G ∈ J(n, Δ)}. In particular, we obtain bounds for b(n, δ_0) and we compute the precise value of a(n, δ_0), α(n, Δ) andβ(n, Δ) for all values of n, δ_0 and Δ, respectively.     

Generalized 3-edge-connectivity of Cartesian product graphs

The generalized k-connectivity κ _k (G) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λ _k (G) = min{λ(S): S ? V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T ₁, T ₂, …, T _l in G such that S ? V (T _i ) for 1 ? i ? l. In this paper we prove that for any two connected graphs G and H we have λ ₃(G □ H) ? λ ₃(G) + λ ₃(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes.     

Some class 1 graphs on <Emphasis Type="Italic">g</Emphasis><Subscript><Emphasis Type="Italic">c</Emphasis></Subscript>-colorings

An edge-coloring of a graph G is an assignment of colors to all the edges of G. A g _c -coloring of a graph G is an edge-coloring of G such that each color appears at each vertex at least g(v) times. The maximum integer k such that G has a g _c -coloring with k colors is called the g _c -chromatic index of G and denoted by \(\chi\prime_{g_{c}}\)(G). In this paper, we extend a result on edge-covering coloring of Zhang and Liu in 2011, and give a new sufficient condition for a simple graph G to satisfy \(\chi\prime_{g_{c}}\)(G) = δ _g (G), where \(\delta_{g}\left(G\right) = min_{v\epsilon V (G)}\left\{\lfloor\frac{d\left(v\right)}{g\left(v\right)}\rfloor\right\}\).     

Graphs with small diameter determined by their <Emphasis Type="Italic">D</Emphasis>-spectra

Let G be a connected graph with vertex set V(G) = {v₁, v₂,..., v _n }. The distance matrix D(G) = (d _ij )_n×n is the matrix indexed by the vertices of G, where d _ij denotes the distance between the vertices v _i and v _j . Suppose that λ₁(D) ≥ λ₂(D) ≥... ≥ λ _n (D) are the distance spectrum of G. The graph G is said to be determined by its D-spectrum if with respect to the distance matrix D(G), any graph having the same spectrum as G is isomorphic to G. We give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their D-spectra.     

Restrained Domination in Claw-Free Graphs with Minimum Degree at Least Two

Let G = (V, E) be a graph. A set \({S\subseteq V}\) is a restrained dominating set if every vertex in V ? S is adjacent to a vertex in S and to a vertex in V ? S. The restrained domination number of G, denoted γ _r (G), is the smallest cardinality of a restrained dominating set of G. We will show that if G is claw-free with minimum degree at least two and \({G\notin \{C_{4},C_{5},C_{7},C_{8},C_{11},C_{14},C_{17}\}}\) , then \({\gamma_{r}(G)\leq \frac{2n}{5}.}\)     

Neighborhood Conditions for Fractional ID-<Emphasis Type="Italic">k</Emphasis>-factor-critical Graphs

Let G be a graph and k ≥ 2 a positive integer. Let h: E(G) → [0, 1] be a function. If \(\sum\limits_{e \mathrel\backepsilon x} {h(e) = k} \) holds for each x ∈ V (G), then we call G[F_h] a fractional k-factor of G with indicator function h where F_h = {e ∈ E(G): h(e) > 0}. A graph G is fractional independent-set-deletable k-factor-critical (in short, fractional ID-k-factor-critical), if G ? I has a fractional k-factor for every independent set I of G. In this paper, we prove that if n ≥ 9k ? 14 and for any subset X ? V (G) we have

$${N_G}(X) = V(G)if|X| \geqslant \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ;or|{N_G}(X)| \geqslant \frac{{3k - 1}}{k}|X|if|X| < \left\lfloor {\frac{{kn}}{{3k - 1}}} \right\rfloor ,$$

then G is fractional ID-k-factor-critical.

     

Constructing New Braided <Emphasis Type="Italic">T</Emphasis>-Categories over Weak Hopf Algebras

Let Aut _{weak Hopf} (H) denote the set of all automorphisms of a weak Hopf algebra H with bijective antipode in the sense of Böhm et al. (J Algebra 221:385–438, 1999) and let G be a certain crossed product group Aut _{weak Hopf} (H)×Aut _{weak Hopf} (H). The main purpose of this paper is to provide further examples of braided T-categories in the sense of Turaev (1994, 2008). For this, we first introduce a class of new categories \( _{H}{\mathcal {WYD}}^{H}(\alpha, \beta)\) of weak (α, β)-Yetter-Drinfeld modules with α, β?∈?Aut _{weak Hopf} (H) and we show that the category \({\mathcal WYD}(H) =\{{}_{H}\mathcal {WYD}^{H}(\alpha, \beta)\}_{(\alpha , \beta )\in G}\) becomes a braided T-category over G, generalizing the main constructions by Panaite and Staic (Isr J Math 158:349–365, 2007). Finally, when H is finite-dimensional we construct a quasitriangular weak T-coalgebra WD(H)?=?{WD(H)_{(α, β)}}_{(α, β)?∈?G} in the sense of Van Daele and Wang (Comm Algebra, 2008) over a family of weak smash product algebras \(\{\overline{H^{*cop}\# H_{(\alpha,\beta)}}\}_{(\alpha , \beta)\in G}\), and we obtain that \({\mathcal {WYD}}(H)\) is isomorphic to the representation category of the quasitriangular weak T-coalgebra WD(H).     

On the rank of compact <Emphasis Type="Italic">p</Emphasis>-adic Lie groups

The rank of a profinite group G is the basic invariant \({{\rm rk}(G):={\rm sup}\{d(H) \mid H \leq G\}}\), where H ranges over all closed subgroups of G and d(H) denotes the minimal cardinality of a topological generating set for H. A compact topological group G admits the structure of a p-adic Lie group if and only if it contains an open pro-p subgroup of finite rank. For every compact p-adic Lie group G one has rk(G) ≥ dim(G), where dim(G) denotes the dimension of G as a p-adic manifold. In this paper we consider the converse problem, bounding rk(G) in terms of dim(G). Every profinite group G of finite rank admits a maximal finite normal subgroup, its periodic radical π(G). One of our main results is the following. Let G be a compact p-adic Lie group such that π(G) = 1, and suppose that p is odd. If \(\{g \in G \mid g^{p-1}=1 \}\) is equal to {1}, then rk(G) = dim(G).     

Unicyclic graphs with bicyclic inverses

A graph is nonsingular if its adjacency matrix A(G) is nonsingular. The inverse of a nonsingular graph G is a graph whose adjacency matrix is similar to A(G)^?1 via a particular type of similarity. Let H denote the class of connected bipartite graphs with unique perfect matchings. Tifenbach and Kirkland (2009) characterized the unicyclic graphs in H which possess unicyclic inverses. We present a characterization of unicyclic graphs in H which possess bicyclic inverses.     

<Emphasis Type="Italic">r</Emphasis>-Dynamic Chromatic Number of Some Line Graphs

An r-dynamic coloring of a graph G is a proper coloring c of the vertices such that |c(N(v))| ≥ min {r, deg(v)}, for each v ∈ V (G). The r-dynamic chromatic number of a graph G is the smallest k such that G admits an r-dynamic coloring with k colors. In this paper, we obtain the r-dynamic chromatic number of the line graph of helm graphs H_n for all r between minimum and maximum degree of H_n. Moreover, our proofs are constructive, what means that we give also polynomial time algorithms for the appropriate coloring. Finally, as the first, we define an equivalent model for edge coloring.     

On Eccentric Connectivity Index of Eccentric Graph of Regular Dendrimer

The eccentric connectivity index \(\xi ^c(G)\) of a connected graph G is defined as \(\xi ^c(G) =\sum _{v \in V(G)}{deg(v) e(v)},\) where deg(v) is the degree of vertex v and e(v) is the eccentricity of v. The eccentric graph, \(G_e\), of a graph G has the same set of vertices as G,  with two vertices u, v adjacent in \(G_e\) if and only if either u is an eccentric vertex of v or v is an eccentric vertex of u. In this paper, we obtain a formula for the eccentric connectivity index of the eccentric graph of a regular dendrimer. We also derive a formula for the eccentric connectivity index for the second iteration of eccentric graph of regular dendrimer.     

<Emphasis Type="Italic">s</Emphasis>-Vertex Pancyclic Index

A graph G is vertex pancyclic if for each vertex \({v \in V(G)}\) , and for each integer k with 3 ≤ k ≤ |V(G)|, G has a k-cycle C _k such that \({v \in V(C_k)}\) . Let s ≥ 0 be an integer. If the removal of at most s vertices in G results in a vertex pancyclic graph, we say G is an s-vertex pancyclic graph. Let G be a simple connected graph that is not a path, cycle or K _1,3. Let l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K ₃}, where a divalent path in G is a path whose interval vertices have degree two in G. The s-vertex pancyclic index of G, written vp _s (G), is the least nonnegative integer m such that L ^m (G) is s-vertex pancyclic. We show that for a given integer s ≥ 0,

$vp_s(G)\le \left\{\begin{array}{l@{\quad}l}\qquad\quad\quad\,\,\,\,\,\,\, l(G)+s+1: \quad {\rm if} \,\, 0 \le s \le 4 \\ l(G)+\lceil {\rm log}_2(s-2) \rceil+4: \quad {\rm if} \,\, s \ge 5 \end{array}\right.$

And we improve the bound for essentially 3-edge-connected graphs. The lower bound and whether the upper bound is sharp are also discussed.

     

Diameter-Preserving Spanning Trees in Sparse Weighted Graphs

We give a simple sufficient condition for a weighted graph to have a diameter-preserving spanning tree. More precisely, let G = (V, E, f _E ) be a connected edge weighted graph with f _E being the edge weight function. Let f _V be the vertex weight function of G induced by f _E as follows: f _V (v) = max{f _E (e) : e is incident with v} for all \({v \in V}\) . We show that G contains a diameter-preserving spanning tree if \({d(G)\ge \frac{2}{3} \sum_{v\in V} f_V(v)}\) where d(G) is the diameter of G. The condition is sharp in the sense that for any \({\epsilon >0 }\) , there exist weighted graphs G satisfying \({d(G) > (\frac{2}{3}-\epsilon)\sum_{v\in V} f_V(v)}\) and not containing a diameter-preserving spanning tree.     

Independence numbers of random subgraphs of distance graphs

We consider the distance graph G(n, r, s), whose vertices can be identified with r-element subsets of the set {1, 2,..., n}, two arbitrary vertices being joined by an edge if and only if the cardinality of the intersection of the corresponding subsets is s. For s = 0, such graphs are known as Kneser graphs. These graphs are closely related to the Erd?s–Ko–Rado problem and also play an important role in combinatorial geometry and coding theory. We study some properties of random subgraphs of G(n, r, s) in the Erd?s–Rényi model, in which every edge occurs in the subgraph with some given probability p independently of the other edges. We find the asymptotics of the independence number of a random subgraph of G(n, r, s) for the case of constant r and s. The independence number of a random subgraph is Θ(log₂n) times as large as that of the graph G(n, r, s) itself for r ≤ 2s + 1, while for r > 2s + 1 one has asymptotic stability: the two independence numbers asymptotically coincide.     

Hopf Modules and Representations of Finite Wreath Products

For a finite group G and nonnegative integer n ≥ 0, one may consider the associated tower \(G \wr S_{n} := S_{n} \ltimes G^{n}\) of wreath product groups. Zelevinsky associated to such a tower the structure of a positive self-adjoint Hopf algebra (PSH-algebra) R(G) on the direct sum over integers n ≥ 0 of the Grothendieck groups K ₀(R e p?G?S _n ). In this paper, we study the interaction via induction and restriction of the PSH-algebras R(G) and R(H) associated to finite groups H ? G. A class of Hopf modules over PSH-algebras with a compatibility between the comultiplication and multiplication involving the Hopf k ^{t h} -power map arise naturally and are studied independently. We also give an explicit formula for the natural PSH-algebra morphisms R(H) → R(G) and R(G) → R(H) arising from induction and restriction. In an appendix, we consider a family of subgroups of wreath product groups analogous to the subgroups G(m, p, n) of the wreath product cyclotomic complex reflection groups G(m, 1, n).     

Some sufficient conditions for Hamiltonian property in terms of Wiener-type invariants

The Wiener-type invariants of a simple connected graph G = (V, E) can be expressed in terms of the quantities \(W_{f}=\sum_{\{u,v\}\subseteq V}f(d_{G}(u,v))\) for various choices of the function f(x), where d_G(u,v) is the distance between vertices u and v in G. In this paper, we give some sufficient conditions for a connected graph to be Hamiltonian, a connected graph to be traceable, and a connected bipartite graph to be Hamiltonian in terms of the Wiener-type invariants.