Bicyclic graphs with exactly two main eigenvalues

An eigenvalue of a graph G is called a main eigenvalue if it has an eigenvector the sum of whose entries is not equal to zero, and it is well known that a graph has exactly one main eigenvalue if and only if it is regular. In this work, all connected bicyclic graphs with exactly two main eigenvalues are determined.     

Mixed Series of Chebyshev Polynomials Orthogonal on a Uniform Grid

We construct an expansion of a discrete function in the form of a mixed series of Chebyshev polynomials. We obtain estimates of the approximation error of the function and its derivatives.     

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.     

Bipartite graphs with the maximum sum of squares of degrees

In this paper we determine all the bipartite graphs with the maximum sum of squares of degrees among the ones with a given number of vertices and edges.     

On the Location of Roots of Graph Polynomials

Roots of graph polynomials such as the characteristic polynomial, the chromatic polynomial, the matching polynomial, and many others are widely studied. In this paper we examine to what extent the location of these roots reflects the graph theoretic properties of the underlying graph.     

第二类Chebyshev多项式系数的绝对值和的卷积

本文主要的目的是利用广义Fibonacci多项式的生成函数及其偏导数来研究第二类Cheby-shev多项式卷积的计算,并给出—个有趣的计算公式.     

All hypoenergetic graphs with maximum degree at most 3

The energy E(G) of a graph G is defined as the sum of the absolute values of its eigenvalues. A connected graph G of order n is said to be hypoenergetic if E(G)<n. All connected hypoenergetic graphs with maximum degree Δ3 have been characterized. In addition to the four (earlier known) hypoenergetic trees, we now show that complete bipartite graph K_2,3 is the only hypoenergetic cycle-containing hypoenergetic graph. By this, the validity of a conjecture by Majstorović et al. has been confirmed.     

Energy and NEPS of graphs

The energy of a graph is the sum of the absolute values of the eigenvalues of the graph. We study the energy of the noncomplete extended p-sum (NEPS) of the graphs, a very general composition of the graphs in which the special case is the product of graphs. We show that the energy of the product of graphs is the product of the energy of graphs, and how this result may be used to construct arbitrarily large families of noncospectral connected graphs having the same number of vertices and the same energy. Further, unlike the product, we show that the energy of any other NEPS of the graphs cannot be represented as a function of the energy of starting graphs.     

There are only finitely many distance-regular graphs of fixed valency greater than two

In this paper we prove the Bannai–Ito conjecture, namely that there are only finitely many distance-regular graphs of fixed valency greater than two.     

Distant total irregularity strength of graphs via random vertex ordering

Let $c:V\cup E\to \{1,2,\dots ,k\}$ be a (not necessarily proper) total colouring of a graph $G=(V,E)$ with maximum degree $\Delta$. Two vertices $u,v\in V$ are sum distinguished if they differ with respect to sums of their incident colours, i.e. $c\left(u\right)+{\sum }_{e?u}c\left(e\right)\ne c\left(v\right)+{\sum }_{e?v}c\left(e\right)$. The least integer $k$ admitting such colouring $c$ under which every $u,v\in V$ at distance $1\le d(u,v)\le r$ in $G$ are sum distinguished is denoted by ${\mathrm{ts}}_r\left(G\right)$. Such graph invariants link the concept of the total vertex irregularity strength of graphs with so-called 1-2-Conjecture, whose concern is the case of $r=1$. Within this paper we combine probabilistic approach with purely combinatorial one in order to prove that ${\mathrm{ts}}_r\left(G\right)\le (2+o\left(1\right)){\Delta }^{r?1}$ for every integer $r\ge 2$ and each graph $G$, thus improving the previously best result: ${\mathrm{ts}}_r\left(G\right)\le 3{\Delta }^{r?1}$.     

On total 9‐coloring planar graphs of maximum degree seven

Given a graph G, a total k‐coloring of G is a simultaneous coloring of the vertices and edges of G with at most k colors. If Δ(G) is the maximum degree of G, then no graph has a total Δ‐coloring, but Vizing conjectured that every graph has a total (Δ + 2)‐coloring. This Total Coloring Conjecture remains open even for planar graphs. This article proves one of the two remaining planar cases, showing that every planar (and projective) graph with Δ ≤ 7 has a total 9‐coloring by means of the discharging method. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 67–73, 1999     

Integral Representations of the Wilson Polynomials and the Continuous Dual Hahn Polynomials

We give representations of the Wilson polynomials and the continuous dual Hahn polynomials in terms of multidimensional generalizations of Barnes type integrals. Motivation is to study the Barnes type integrals from the viewpoint of the de Rham theory and holonomic systems.     

A New Integral Involving the Product of Bessel Functions and Associated Laguerre Polynomials

A new result for integrals involving the product of Bessel functions and Associated Laguerre polynomials is obtained in terms of the hypergeometric function. Some special cases of the general integral lead to interesting finite and infinite series representations of hypergeometric functions.     

Integer Roots Chromatic Polynomials of Non-Chordal Graphs and the Prouhet-Tarry-Escott Problem

In this paper, we give an affirmative answer to a question of Dmitriev concerning the existence of a non-chordal graph with a chordless cycle of order n whose chromatic polynomial has integer roots for a few values of n, extending prior work of Dong et al. Received: April, 2003     

The number of edges of radius-invariant graphs

The eccentricity e(υ) of vertex υ is defined as a distance to a farthest vertex from υ. The radius of a graph G is defined as r(G) = {e(u)}. We consider properties of unchanging the radius of a graph under two different situations: deleting an arbitrary edge and deleting an arbitrary vertex. This paper gives the upper bounds for the number of edges in such graphs. Supported by VEGA grant No. 1/0084/08.     

Commuting decompositions of complete graphs

We say that two graphs G and H with the same vertex set commute if their adjacency matrices commute. In this article, we show that for any natural number r, the complete multigraph K is decomposable into commuting perfect matchings if and only if n is a 2‐power. Also, it is shown that the complete graph K_n is decomposable into commuting Hamilton cycles if and only if n is a prime number. © 2006 Wiley Periodicals, Inc. J Combin Designs