Solution to a conjecture on the maximal energy of bipartite bicyclic graphs

The energy of a simple graph G, denoted by E(G), is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let C_n denote the cycle of order n and the graph obtained from joining two cycles C₆ by a path P_n-12 with its two leaves. Let B_n denote the class of all bipartite bicyclic graphs but not the graph R_a,b, which is obtained from joining two cycles C_a and C_b (a,b≥10 and ) by an edge. In [I. Gutman, D. Vidovi?, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41(2001) 1002-1005], Gutman and Vidovi? conjectured that the bicyclic graph with maximal energy is , for n=14 and n≥16. In [X. Li, J. Zhang, On bicyclic graphs with maximal energy, Linear Algebra Appl. 427(2007) 87-98], Li and Zhang showed that the conjecture is true for graphs in the class B_n. However, they could not determine which of the two graphs R_a,b and has the maximal value of energy. In [B. Furtula, S. Radenkovi?, I. Gutman, Bicyclic molecular graphs with the greatest energy, J. Serb. Chem. Soc. 73(4)(2008) 431-433], numerical computations up to a+b=50 were reported, supporting the conjecture. So, it is still necessary to have a mathematical proof to this conjecture. This paper is to show that the energy of is larger than that of R_a,b, which proves the conjecture for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, the conjecture is still open.     

On bicyclic graphs with maximal energy

Let λ₁,λ₂,…,λ_n be the eigenvalues of a graph G of order n. The energy of G is defined as E(G)=|λ₁|+|λ₂|+?+|λ_n|. Let be the graph obtained from two copies of C₆ joined by a path P_n-10, B_n be the class of all bipartite bicyclic graphs that are not the graph obtained from two cycles C_a and C_b (a,b?10 and a≡b≡2 (mod 4)) joined by an edge. In this paper, we show that is the graph with maximal energy in B_n, which gives a partial solution to Gutman’s conjecture in Gutman and Vidovi? (2001) [I. Gutman, D. Vidovi?, Quest for molecular graphs with maximal energy: a computer experiment, J. Chem. Inf. Sci. 41 (2001) 1002-1005].     

The total graph and regular graph of a commutative ring

Let R be a commutative ring. The total graph of R, denoted by T(Γ(R)) is a graph with all elements of R as vertices, and two distinct vertices x,y∈R, are adjacent if and only if x+y∈Z(R), where Z(R) denotes the set of zero-divisors of R. Let regular graph of R, Reg(Γ(R)), be the induced subgraph of T(Γ(R)) on the regular elements of R. Let R be a commutative Noetherian ring and Z(R) is not an ideal. In this paper we show that if T(Γ(R)) is a connected graph, then . Also, we prove that if R is a finite ring, then T(Γ(R)) is a Hamiltonian graph. Finally, we show that if S is a commutative Noetherian ring and Reg(S) is finite, then S is finite.     

The energy of unitary cayley graphs

A graph G of order n is called hyperenergetic if E(G)>2n-2, where E(G) denotes the energy of G. The unitary Cayley graph X_n has vertex set Z_n={0,1,2,…,n-1} and vertices a and b are adjacent, if gcd(a-b,n)=1. These graphs have integral spectrum and play an important role in modeling quantum spin networks supporting the perfect state transfer. We show that the unitary Cayley graph X_n is hyperenergetic if and only if n has at least two prime factors greater than 2 or at least three distinct prime factors. In addition, we calculate the energy of the complement of unitary Cayley graph and prove that is hyperenergetic if and only if n has at least two distinct prime factors and n≠2p, where p is a prime number. By extending this approach, for every fixed , we construct families of k hyperenergetic non-cospectral integral circulant n-vertex graphs with equal energy.     

Asymmetric marking games on line graphs

This paper investigates the asymmetric marking games on line graphs. Suppose G is a graph with maximum degree Δ and G has an orientation with maximum outdegree k, we show that the (a,1)-game coloring number of the line graph of G is at most . When a=1, this improves some known results of the game coloring number of the line graphs.     

More on decompositions of edge-colored complete graphs

Let G be a family of graphs whose edges are colored with elements from a set R of r colors. We assume no two vertices of G are joined by more than one edge of color i for any i∈R, for each G∈G. will denote the complete graph with r edges joining any pair of distinct vertices, one of each of the r colors. We describe necessary and asymptotically sufficient conditions on n for the existence of a family D of subgraphs of , each of which is an isomorphic copy of some graph in G, so that each edge of appears in exactly one of the subgraphs in D.     

Homoclinic solutions for a class of the second order Hamiltonian systems

We study the existence of homoclinic orbits for the second order Hamiltonian system , where q∈Rⁿ and V∈C¹(R×Rⁿ,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b₁|q|²?K(t,q)?b₂|q|², W is superlinear at the infinity and f is sufficiently small in L²(R,Rⁿ). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations.     

Heteroclinic solutions for a class of the second order Hamiltonian systems

We shall be concerned with the existence of heteroclinic orbits for the second order Hamiltonian system , where q∈Rn and V∈C¹(R×Rn,R), V?0. We will assume that V and a certain subset M⊂Rn satisfy the following conditions. M is a set of isolated points and #M?2. For every sufficiently small ε>0 there exists δ>0 such that for all (t,z)∈R×Rn, if d(z,M)?ε then −V(t,z)?δ. The integrals , z∈M, are equi-bounded and −V(t,z)→∞, as |t|→∞, uniformly on compact subsets of Rn?M. Our result states that each point in M is joined to another point in M by a solution of our system.     

Topology of dynamical systems in finite groups and number theory

We study and develop a very new object introduced by V.I. Arnold: a monad is a triple consisting of a finite set, a map from that finite set to itself and the monad graph which is the directed graph whose vertices are the elements of the finite set and whose arrows lead each vertex to its image (by the map). We consider the case in which the finite set entering in the monad definition is a finite group G and the map is the Frobenius map, for some k∈Z. We study the Frobenius dynamical system defined by the iteration of the monad fk, and also study the combinatorics and topology (i.e., the discrete invariants) of the monad graph. Our study provides useful information about several structures on the group associated to the monad graph. So, for example, several properties of the quadratic residues of finite commutative groups can be obtained in terms of the graph of the Frobenius monad .     

Density of constant radius normal binary covering codes

A binary code with covering radius R is a subset C of the hypercube Q_n={0,1}ⁿ such that every x∈Q_n is within Hamming distance R of some codeword c∈C, where R is as small as possible. For a fixed coordinate i∈[n], define to be the set of codewords with a b in the ith position. Then C is normal if there exists an i∈[n] such that for any v∈Q_n, the sum of the Hamming distances from v to and is at most 2R+1. We newly define what it means for an asymmetric covering code to be normal, and consider the worst-case asymptotic densities ν^*(R) and of constant radius R symmetric and asymmetric normal covering codes, respectively. Using a probabilistic deletion method, and analysis adapted from previous work by Krivelevich, Sudakov, and Vu, we show that and , giving evidence that minimum size constant radius covering codes could still be normal.     

4-regular claw-free IM-extendable graphs

A graph G is induced matching extendable (shortly, IM-extendable), if every induced matching of G is included in a perfect matching of G. A graph G is claw-free, if G does not contain any induced subgraph isomorphic to K_1,3. The kth power of a graph G, denoted by G^k, is the graph with vertex set V(G) in which two vertices are adjacent if and only if the distance between them in G is at most k. In this paper, the 4-regular claw-free IM-extendable graphs are characterized. It is shown that the only 4-regular claw-free connected IM-extendable graphs are , and T_r, r?2, where T_r is the graph with 4r vertices u_i,v_i,x_i,y_i, 1?i?r, such that for each i with 1?i?r, {u_i,v_i,x_i,y_i} is a clique of T_r and . We also show that a 4-regular strongly IM-extendable graph must be claw-free. As a consequence, the only 4-regular strongly IM-extendable graphs are K₄×K₂, and .     

Dense edge-magic graphs and thin additive bases

A graph G of order n and size m is edge-magic if there is a bijection l:V(G)∪E(G)→[n+m] such that all sums l(a)+l(b)+l(ab), ab∈E(G), are the same. We present new lower and upper bounds on M(n), the maximum size of an edge-magic graph of order n, being the first to show an upper bound of the form . Concrete estimates for ε can be obtained by knowing s(k,n), the maximum number of distinct pairwise sums that a k-subset of [n] can have.So, we also study s(k,n), motivated by the above connections to edge-magic graphs and by the fact that a few known functions from additive number theory can be expressed via s(k,n). For example, our estimate

     

On the diameter and girth of a zero-divisor graph

For a commutative ring R with zero-divisors Z(R), the zero-divisor graph of R is Γ(R)=Z(R)−{0}, with distinct vertices x and y adjacent if and only if xy=0. In this paper, we characterize when either or . We then use these results to investigate the diameter and girth for the zero-divisor graphs of polynomial rings, power series rings, and idealizations.     

Compact almost automorphic solutions to integral equations with infinite delay

Given a∈L¹(R) and the generator A of an L¹-integrable resolvent family of linear bounded operators defined on a Banach space X, we prove the existence of compact almost automorphic solutions of the semilinear integral equation for each f:R×X→X compact almost automorphic in t, for each x∈X, and satisfying Lipschitz and Hölder type conditions. In the scalar linear case, we prove that a∈L¹(R) positive, nonincreasing and log-convex is sufficient to obtain the existence of compact almost automorphic solutions.     

Two spectral characterizations of regular, bipartite graphs with five eigenvalues

Graphs with a few distinct eigenvalues usually possess an interesting combinatorial structure. We show that regular, bipartite graphs with at most six distinct eigenvalues have the property that each vertex belongs to the constant number of quadrangles. This enables to determine, from the spectrum alone, the feasible families of numbers of common neighbors for each vertex with other vertices in its part. For particular spectra, such as [6,2⁹,0⁶,-2⁹,-6] (where exponents denote eigenvalue multiplicities), there is a unique such family, which makes it possible to characterize all graphs with this spectrum.Using this lemma we also to show that, for r?2, a graph has spectrum if and only if it is a graph of a 1-resolvable transversal design TD(r,r), i.e., if it corresponds to the complete set of mutually orthogonal Latin squares of size r in a well-defined manner.     

Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .     

Unprovability threshold for the planar graph minor theorem

This note is part of the implementation of a programme in foundations of mathematics to find exact threshold versions of all mathematical unprovability results known so far, a programme initiated by Weiermann. Here we find the exact versions of unprovability of the finite graph minor theorem with growth rate condition restricted to planar graphs, connected planar graphs and graphs embeddable into a given surface, assuming an unproved conjecture (*): ‘there is a number a>0 such that for all k≥3, and all n≥1, the proportion of connected graphs among unlabelled planar graphs of size n omitting the k-element circle as minor is greater than a’. Let γ be the unlabelled planar growth constant (27.2269≤γ<30.061). Let P(c) be the following first-order arithmetical statement with real parameter c: “for every K there is N such that whenever G₁,G₂,…,G_N are unlabelled planar graphs with |G_i|<K+c⋅log₂i then for some i<j≤N, G_i is isomorphic to a minor of G_j”. Then

1.

for every , P(c) is provable in IΔ₀+exp;

2.

for every , P(c) is unprovable in .

We also give proofs of some upper and lower bounds for unprovability thresholds in the general case of the finite graph minor theorem.     

Generalized Haar integral

The aim of the paper is to generalize the notion of the Haar integral. For a compact semigroup S acting continuously on a Hausdorff compact space Ω, the algebra A(S)⊂C(Ω,R) of S-invariant functions and the linear space M(S) of S-invariant (real-valued) finite signed measures are considered. It is shown that if S has a left and right invariant measure, then the dual space of A(S) is isometrically lattice-isomorphic to M(S) and that there exists a unique linear operator (called the Haar integral) such that for each f∈A(S) and for any f∈C(Ω,R) and s∈S, , where .