Balanced extensions of graphs and hypergraphs

For a hypergraphG withv vertices ande _i edges of sizei, the average vertex degree isd(G)= ∑ie ₁/v. Callbalanced ifd(H)≦d(G) for all subhypergraphsH ofG. Let $$m(G) = \mathop {\max }\limits_{H \subseteqq G} d(H).$$ A hypergraphF is said to be abalanced extension ofG ifG?F, F is balanced andd(F)=m(G), i.e.F is balanced and does not increase the maximum average degree. It is shown that for every hypergraphG there exists a balanced extensionF ofG. Moreover everyr-uniform hypergraph has anr-uniform balanced extension. For a graphG let ext (G) denote the minimum number of vertices in any graph that is a balanced extension ofG. IfG hasn vertices, then an upper bound of the form ext(G) ₁ n ² is proved. This is best possible in the sense that ext(G)>c ₂ n ² for an infinite family of graphs. However for sufficiently dense graphs an improved upper bound ext(G) ₃ n can be obtained, confirming a conjecture of P. Erdõs.     

Eccentric sequences in graphs

The eccentricitye(v) of a vertexv of a connected graphG is the maximum distance fromv among the vertices ofG. A nondecreasing sequencea ₁,a ₂, ...,a _p of nonnegative integers is said to be an eccentric sequence if there exists a connected graphG of orderp whose vertices can be labelledv ₁,v ₂, ...,v _p so thate(v _i )=a _i for alli. Several properties of eccentric sequences are exhibited, and a necessary and sufficient condition for a sequence to be eccentric is presented. Sequences which are the eccentricity sequences of trees are characterized. Some properties of the eccentricity sequences of self-complementary graphs are obtained. It is shown that the radius of a nontrivial self-complementary graph is two.     

Asynchronous threshold networks

LetG=(V,E) be a graph with an initial signs(v)∈{±1} for every vertexv∈V. When a certexv becomesactive, it resets its sign tos′(v) which is the sign of the majority of its neighbors(s′(v)=1 if there is a tie).G is in astable state if,s′(v) for allv∈V. We show that for every graphG=(V,E) and every initial signs, there is a sequencev ₁,v ₂,...,v _r of vertices ofG, in which no vertex appears more than once, such that ifv _i becomes active at timei, (1≤i≤r), then after theser stepsG reaches a stable state. This proves a conjecture of Miller. We also consider some generalizations to directed graphs with weighted edges.     

A relationship between bounds on the sum of squares of degrees of a graph

LetG = (V, E) be a simple graph withn vertices and e edges. Letdi be the degree of the ith vertex v_i ∈ V andm _i the average of the degrees of the vertices adjacent to vertexv _i ∈ V. It is known by Caen [1] and Das [2] that $\frac{{4e^2 }}{n} \leqslant d_1^2 + ... + d_n^2 \leqslant e max \{ d_j + m_j |v_j \in V\} \leqslant e\left( {\frac{{2e}}{{n - 1}} + n - 2} \right)$ . In general, the equalities do not hold in above inequality. It is shown that a graphG is regular if and only if $\frac{{4e^2 }}{n} = d_1^2 + ... + d_n^2 $ . In fact, it is shown a little bit more strong result that a graphG is regular if and only if $\frac{{4e^2 }}{n} = d_1^2 + ... + d_n^2 = e max \{ d_j + m_j |v_j \in V\} $ . For a graphG withn < 2 vertices, it is shown that G is a complete graphK _n if and only if $\frac{{4e^2 }}{n} = d_1^2 + ... + d_n^2 = e max \{ d_j + m_j |v_j \in V\} = e\left( {\frac{{2e}}{{n - 1}} + n - 2} \right)$ .     

An unfeasible matching problem

LetG be a bipartite graph with natural edge weights, and letW be a function from the set of vertices ofG into natural numbers. AW-matching ofG is a subset of the set of edges ofG such that for each vertexv the total weight of edges in the subset incident tov does not exceedW(v). Letm be a natural number. We show that the problem of deciding whether there is aW-matching inG whose total weight is not less thanm is NP-complete even ifG is bipartite and its edge weights as well as theW(v)-constraints are constantly bounded.     

Geodeticity of the contour of chordal graphs

A vertex v is a boundary vertex of a connected graph G if there exists a vertex u such that no neighbor of v is further away from u than v. Moreover, if no vertex in the whole graph V(G) is further away from u than v, then v is called an eccentric vertex of G. A vertex v belongs to the contour of G if no neighbor of v has an eccentricity greater than the eccentricity of v. Furthermore, if no vertex in the whole graph V(G) has an eccentricity greater than the eccentricity of v, then v is called a peripheral vertex of G. This paper is devoted to study these kinds of vertices for the family of chordal graphs. Our main contributions are, firstly, obtaining a realization theorem involving the cardinalities of the periphery, the contour, the eccentric subgraph and the boundary, and secondly, proving both that the contour of every chordal graph is geodetic and that this statement is not true for every perfect graph.     

Some properties onf-edge covered critical graphs

LetG(V, E) be a simple graph, and letf be an integer function onV with 1 ≤f(v) ≤d(v) to each vertexv ∈V. An f-edge cover-coloring of a graphG is a coloring of edge setE such that each color appears at each vertexv ∈V at leastf(v) times. Thef-edge cover chromatic index ofG, denoted by χ′ _fc (G), is the maximum number of colors such that anf-edge cover-coloring ofG exists. Any simple graphG has anf-edge cover chromatic index equal to δ_f or δ _f - 1, where $\delta _f = \mathop {\min }\limits_{\upsilon \in V} \{ \left\lfloor {\frac{{d(v)}}{{f(v)}}} \right\rfloor \} $ . LetG be a connected and not complete graph with χ′ _fc (G)=δ _f-1, if for eachu, v ∈V and e =uv ?E, we have ÷ _fc (G + e) > ÷ _fc (G), thenG is called anf-edge covered critical graph. In this paper, some properties onf-edge covered critical graph are discussed. It is proved that ifG is anf-edge covered critical graph, then for eachu, v ∈V and e =uv ?E there existsw ∈ {u, v } withd(w) ≤ δ _f (f(w) + 1) - 2 such thatw is adjacent to at leastd(w) - δ _f + 1 vertices which are all δ _f -vertex inG.     

Two edge-disjoint hamiltonian cycles in graphs

E. Schmeichel and D. Hayes showed that ifG is a 2-connected graph withd(u) +d(v)≥n ?1 for every pair of nonadjacent vertices andv, then G has a Hamiltonian cycle unlessG is the graph of Fig. 2 (b). In this paper, it is proved that, under almost the same conditions as Schmeichel and Hayes’s Theorem, namely,G is a 2-connected graph of ordern (n ≥ 40) with δ(G) ≥ 7 for every pair of nonadjacent vertices andv, G has two edge-disjoint Hamiltonian cycles unlessG is one of the graphs in Fig. 1 or Fig. 2, and this conclusion is best possible.     

A generalization of a result of Bauer and Schmeichel

We prove that if G is a 1-tough graph withn ≥ 3 vertices such thatd(u) + d(v) +d(w) ≥n+ κ —2 holds for any triple of independent verticesu, v andw ofG, thenG is hamiltonian, wherek is the vertex connectivity ofG. This generalizes a recent result of Baur and Schmeichel.     

(g, f)-factorizations orthogonal to a subgraph in graphs

LetG be a graph with vertex setV (G) and edge setE (G), and letg andf be two integer-valued functions defined on V(G) such thatg(x)⩽(x) for every vertexx ofV(G). It was conjectured that ifG is an (mg +m - 1,mf -m+1)-graph andH a subgraph ofG withm edges, thenG has a (g,f)-factorization orthogonal toH. This conjecture is proved affirmatively. Project supported by the National Natural Science Foundation of China.     

Reconstructing graphs as subsumed graphs of hypergraphs,and some self-complementary triple systems

LetV be a set ofn elements. The set of allk-subsets ofV is denoted . Ak-hypergraph G consists of avertex-set V(G) and anedgeset , wherek≥2. IfG is a 3-hypergraph, then the set of edges containing a given vertexvεV(G) define a graphG _v . The graphs {G _v νvεV(G)} aresubsumed byG. Each subsumed graphG _v is a graph with vertex-setV(G) − v. They can form the set of vertex-deleted subgraphs of a graphH, that is, eachG _v ≽H −v, whereV(H)=V(G). In this case,G is a hypergraphic reconstruction ofH. We show that certain families of self-complementary graphsH can be reconstructed in this way by a hypergraphG, and thatG can be extended to a hypergraphG ^*, all of whose subsumed graphs are isomorphic toH, whereG andG ^* are self-complementary hypergraphs. In particular, the Paley graphs can be reconstructed in this way. This work was supported by an operating grant from the Natural Sciences and Engineering Research Council of Canada.     

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.     

On the power of a perturbation for testing non-isomorphism of graphs

In this paper we prove an inverted version of A. J. Schwenk's result, which in turn is related to Ulam's reconstruction conjecture. Instead of deleting vertices from an undirected graphG, we add a new vertexv and join it to all other vertices ofG to get a perturbed graphG+v. We derive an expression for the characteristic polynomial of the perturbed graphG+v in terms of the characteristic polynomial of the original graphG. We then show the extent to which the characteristic polynomials of the perturbed graphs can be used in determining whether two graphs are non-isomorphic.This work was supported by the U.S. Army Research Office under Grant DAAG29-82-K-0107.     

Degree Distance of Unicyclic Graphs with Given Matching Number

Let G be a connected graph with vertex set V(G). The degree distance of G is defined as ${D'(G) = \sum_{\{u, v\}\subseteq V(G)} (d_G(u) + d_G (v))\, d(u,v)}$ , where d _G (u) is the degree of vertex u, d(u, v) denotes the distance between u and v, and the summation goes over all pairs of vertices in G. In this paper, we characterize n-vertex unicyclic graphs with given matching number and minimal degree distance.     

On products of two nilpotent subgroups of a finite group

LetG be a finite group with an abelian Sylow 2-subgroup. LetA be a nilpotent subgroup ofG of maximal order satisfying class (A)≦k, wherek is a fixed integer larger than 1. Suppose thatA normalizes a nilpotent subgroupB ofG of odd order. ThenAB is nilpotent. Consequently, ifF(G) is of odd order andA is a nilpotent subgroup ofG of maximal order, thenF(G)?A.     

Bounds of eigenvalues of a graph

LetG be a simple graph withn vertices. We denote by _i(G) thei-th largest eigenvalue ofG. In this paper, several results are presented concerning bounds on the eigenvalues ofG. In particular, it is shown that –1₂(G)(n–2)/2, and the left hand equality holds if and only ifG is a complete graph with at least two vertices; the right hand equality holds if and only ifn is even andG2K _n/2.     

Ramsey problem on multiplicities of complete subgraphs in nearly quasirandom graphs

Letk _t(G) be the number of cliques of ordert in the graphG. For a graphG withn vertices let \(c_t (G) = \frac{{k_t (G) + k_t (\bar G)}}{{\left( {\begin{array}{*{20}c} n \\ t \\ \end{array} } \right)}}\) . Letc _t(n)=Min{c _t(G)∶?G?=n} and let \(c_t = \mathop {\lim }\limits_{n \to \infty } c_t (n)\) . An old conjecture of Erdös [2], related to Ramsey's theorem states thatc _t=2^1-(t/2). Recently it was shown to be false by A. Thomason [12]. It is known thatc _t(G)≈2^1-(t/2) wheneverG is a pseudorandom graph. Pseudorandom graphs — the graphs “which behave like random graphs” — were inroduced and studied in [1] and [13]. The aim of this paper is to show that fort=4,c _t(G)≥2^1-(t/2) ifG is a graph arising from pseudorandom by a small perturbation.     

On the greatest number of 2 and 3 colorings of a (v,e)-graph

Let F denote the family of simple undirected graphs on v vertices having e edges ((v, e)-graphs) and P(λ, G) be the chromatic polynomial of a graph G. For the given integers v, e, Δ, let f(v, e, Δ) denote the greatest number of proper colorings in Δ or less colors that a (v, e)-graph G can have, i.e., f(v, e, Δ) = max{P(Δ, G): G ∈ F}. In this paper we determine f(v, e, 2) and describe all graphs G for which P(2, G) = f(v, e, 2). For f(v, e, 3), a lower bound and an upper bound are found.     

On the chromaticity of the 2-degree integral subgraph ofq-trees

A graphG is called to be a 2-degree integral subgraph of aq-tree if it is obtained by deleting an edge e from an integral subgraph that is contained in exactlyq- 1 triangles. An added-vertexq-treeG with n vertices is obtained by taking two verticesu, v (u, v are not adjacent) in a q-treesT withn -1 vertices such that their intersection of neighborhoods ofu, v forms a complete graphK _q , and adding a new vertexx, new edgesxu, xv, xv ₁,xv ₂, …,xv _{q- 4}, where {v ₁,v ₂,...,v _q?4} ?-K _q . In this paper we prove that a graphG with minimum degree not equal toq -3 and chromatic polynomialP(G;λ) = λ(λ - 1) … (λ -q +2)(λ -q +1)³(λ -q) ^{n- q- 2} withn ≥ q + 2 has and only has 2-degree integral subgraph of q-tree withn vertices and added-vertex q-tree withn vertices.