Polynomial Of Graph Research Articles

The Eccentric-Distance Sum Polynomials of Graphs by Using Graph Products

The correlations between the physico-chemical properties of a chemical structure and its molecular structure-properties are used in quantitative structure-activity and property relationship studies (QSAR/QSPR) by using graph-theoretical analysis and techniques. It is well known that some structure-activity and quantitative structure-property studies, using eccentric distance sum, are better than the corresponding values obtained by using the Wiener index. In this article, we give precise expressions for the eccentric distance sum polynomial of some graph products such as join, Cartesian, lexicographic, corona and generalized hierarchical products. We implement our outcomes to calculate this polynomial for some significant families of molecular graphs in the form of the above graph products.

Quantum walks do not like bridges

We consider graphs with two cut vertices joined by a path with one or two edges, and prove that there can be no quantum perfect state transfer between these vertices, unless the graph has no other vertex. We achieve this result by applying the 1-sum lemma for the characteristic polynomial of graphs, the neutrino identities that relate entries of eigenprojectors and eigenvalues, and variational principles for eigenvalues (Cauchy interlacing theorem, Weyl inequalities and Wielandt minimax principle). We see our result as an intermediate step to broaden the understanding of how connectivity plays a key role in quantum walks, and as further evidence of the conjecture that no tree on four or more vertices admits state transfer. We conclude with some open problems.

Generalized spectral characterization of rooted product graphs

A graph G is said to be determined by its generalized spectrum (DGS for short) if for any graph H, whenever H and G are cospectral with cospectral complements, then H is isomorphic to G. In Mao et al. [A new method for constructing graphs determined by their generalized spectrum. Linear Algebra Appl. 2015;477:112–127.], the authors gave an explicit method of constructing large DGS graphs from small ones. More precisely, let G be a graph on n vertices with adjacency matrix A, and (e denotes the all-one vector) be its walk-matrix. The authors proved, among others, that the rooted product graph (i.e. a graph obtained from G by adding a pendent edge at every vertex of G) is DGS whenever (n is even) and the constant term of the characteristic polynomial of graph G is . In this paper, we shall extend the above result in two directions, namely, in the above-rooted graph , we may replace the pendent edge with a path for k = 3 or 4; we are also able to enlarge the family of graphs G satisfying to a family of graphs , in which every graph G satisfies that is odd and square-free. We prove that, under certain conditions, the new rooted product graphs are still DGS. In particular, based on the above result we obtain infinite sequences of DGS graphs for some DGS graph G.

A Deletion–Contraction Relation for the DP Color Function

DP-coloring is a generalization of list coloring that was introduced in 2015 by Dvořák and Postle. The chromatic polynomial of a graph G, denoted P(G, m), is equal to the number of proper m-colorings of G. A well-known tool for computing the chromatic polynomial of graph G is the deletion-contraction formula which relates P(G, m) to the chromatic polynomials of two smaller graphs. The DP color function of a graph G, denoted $$P_{DP}(G,m)$$ , is a DP-coloring analogue of the chromatic polynomial, and $$P_{DP}(G,m)$$ is the minimum number of DP-colorings of G over all possible m-fold covers. In this paper we present a deletion-contraction relation for the DP color function. To make this possible, we extend the definition of the DP color function to multigraphs. We also introduce the dual DP color function of a graph G, denoted $$P^*_{DP}(G,m)$$ , which counts the maximum number of DP-colorings of G over certain m-fold covers. We show how the dual DP color function along with our deletion-contraction relation yields a new general lower bound on the DP color function of a graph.

On counting polynomials of certain classes of polycyclic aromatic hydrocarbons

Polya was the first to establish the concept of a counting polynomial in chemistry. Despite the fact that the spectra of the characteristic polynomial of graphs were exhaustively researched by quantitative means in order to ascertain the molecular orbitals of unsaturated hydrocarbons, the field garnered little attention from chemists for several decades. Polynomials can be used to produce a variety of significant topological indices, either explicitly or after calculating derivatives or numerical methods. Certain firmly related polynomials, namely Theta, Omega, PI, and Sadhana polynomials, subordinate the equidistant edges and non-equidistant edges of structural graphs, which have a rich application in computing the corresponding topological indices. A topological index is a numerical value associated with a network that predicts chemical, physical, and biological activities. This article investigates these polynomials for three classes of polycyclic aromatic hydrocarbons as hexa-peri-hexabenzocoronene, hexa-cata-hexabenzocoronene, and dodeca-benzo-circumcoronene. In addition, closed formulas for these polynomials’ corresponding indices are proposed.

Chromatic Polynomial of Intuitionistic Fuzzy Graphs Using α , β -Levels

The article describes a new thought on the chromatic polynomial of an intuitionistic fuzzy graph which is illustrated based on α , β -level graphs. Besides, the alpha-beta fundamental set of an intuitionistic fuzzy graph is also defined with a vivid description. In addition to that, some characterizations of the chromatic polynomial of an intuitionistic fuzzy graph are specified as well verified. Furthermore, some untouched properties of the α , β -level graph are also projected and proved.

2-domination polynomials of graphs

Let G be a simple graph of order m. Let D2(G, i) be the family of 2-dominating sets in G with size i. The polynomial D2(G, ) = is called the 2-domination polynomial of G. Let D2(Sm, i) be the family of 2-dominating sets of the spider graph Sm with cardinality i and let d2(Sm, i) = |D2(Sm, i)|. Then the 2-domination polynomial D2(Sm, ) of Sm is defined as D2(Sm, ) = , where is the 2-domination number of Sm. In this paper, we obtain some operations on graphs.

Independent domination polynomial of zero-divisor graphs of commutative rings

Independent domination polynomial of zero-divisor graphs of commutative rings

A Categorification for the Signed Chromatic Polynomial

By coloring a signed graph by signed colors, one obtains the signed chromatic polynomial of the signed graph. For each signed graph we construct graded cohomology groups whose graded Euler characteristic yields the signed chromatic polynomial of the signed graph. We show that the cohomology groups satisfy a long exact sequence which categorifies the signed deletion-contraction rule. This work is motivated by Helme-Guizon and Rong's construction of the categorification for the chromatic polynomial of unsigned graphs.

Diverse-Region Hyperspectral Image Classification via Superpixelwise Graph Convolution Technique

In this paper, a diverse-region hyperspectral image classification (DRHy) method is proposed by considering both irregularly local pixels and globally contextual connections between pixels. Specifically, the proposed method is operated on non-Euclidean graphs, which are constructed by superpixel segmentation methods for diverse regions to cluster irregularly local-region pixels. In addition, the dimensionality reduction method is employed to alleviate the curse of dimensionality problem with a lower computational burden, generating more representative data with the input graph features. In this context, it then constructs a superpixelwise Chebyshev polynomial graph convolution network (ChebyNet) to aggregate global-region superpixels. Benefiting from different superpixel numbers of segmentations, we construct different graph structures, and multiple classification results are obtained, which brings more opportunities to represent the hyperspectral data correctly. Then, all the diverse-region results are further fused by a majority voting technique to improve the final performance. Finally, numerical experiments on two benchmark datasets are provided to demonstrate the superiority of the proposed DRHy-ChebyNet method to the other state-of-the-art methods.

The Tutte polynomial of phenylene systems with given number of branching hexagons

AbstractPolynomial graph invariants have been confirmed to have important applications in quantum chemistry and biological information. One of the famous polynomial graph invariants is the Tutte polynomial which gives abundant graph‐theoretical information of the underlying graph. In this paper, we first give a simpler and more efficient method to get the Tutte polynomials of alternating polycyclic chains. Then we obtain the explicit expressions for the Tutte polynomials and the number of spanning trees of phenylene systems with given number of branching hexagons. Moreover, we determine the extremal values of the number of spanning trees among the phenylene systems with given one branching hexagon and two branching hexagons. The corresponding extremal phenylene systems are characterized, respectively.

Independence Polynomials of Bipartite Graphs

Independence Polynomials of Bipartite Graphs

Peripheral Hosoya polynomial of composite graphs

Peripheral Hosoya polynomial of composite graphs

Design of Measurement of Water Content with Capacity Method to Determine Old Categories to Save Tapioca Flour

In this study, water content measurements were carried out using a physical method using a capacitive sensor. Measurement with these two methods aims to get the most efficient method for tapioca flour's water content value. The samples used were six samples of tapioca flour from the way good factory. Each sample was measured using a thermogravimetric method and a capacitive sensor measuring instrument three times per sample. The measurement results produce a polynomial graph with a coefficient of determination of 0.9702. The graph shows that the higher the water content, the higher the value detected by the capacitive sensor. The thermogravimetric method used as a reference takes 6 hours to get results, and the physical method takes less than one minute. To compare measurements, measurements were made using the thermogravimetric method to measure the percentage of water content in tapioca flour. This study indicates that the physical measurement method with a capacitive sensor instrument requires a shorter time than the thermogravimetric method. And the higher the water content, the shorter the storage time.

Spectral Invariants and Their Application on Spectral Characterization of Graphs

In this paper, we give a method to characterize graphs determined by their adjacency spectrum. At first, we give two parameters Π1(G) and Π2(G), which are related to coefficients of the characteristic polynomial of graph G. All connected graphs with Π1(G)∈{1,0,−1,−2,−3} and Π2(G)∈{0,−1,−2,−3} are characterized. Some interesting properties of Π1(G) and Π2(G) are also given. We then find the necessary and sufficient conditions for two classes of graphs to be determined by their adjacency spectrum.

On the Unimodality of Domination Polynomials

A polynomial is said to be unimodal if its coefficients are non-decreasing and then non-increasing. The domination polynomial of a graph G is the generating function of the number of dominating sets of each cardinality in G, and its coefficients have been conjectured to be unimodal. In this paper we will show the domination polynomial of paths, cycles and complete multipartite graphs are unimodal, and that the domination polynomial of almost every graph is unimodal with mode $$ \lceil \frac{n}{2}\rceil $$ .

The Neighborhood Polynomial of Chordal Graphs

We study the neighborhood polynomial and the complexity of its computation for chordal graphs. The neighborhood polynomial of a graph is the generating function of subsets of its vertices that have a common neighbor. We introduce a parameter for chordal graphs called anchor width and an algorithm to compute the neighborhood polynomial which runs in polynomial time if the anchor width is polynomially bounded. The anchor width is the maximal number of different sub-cliques of a clique which appear as a common neighborhood. Furthermore we study the anchor width for chordal graphs and some subclasses such as chordal comparability graphs and chordal graphs with bounded leafage. the leafage of a chordal graphs is the minimum number of leaves in the host tree of a subtree representation. We show that the anchor width of a chordal graph is at most $n^{\ell}$ where $\ell$ denotes the leafage. This shows that for some subclasses computing the neighborhood polynomial is possible in polynomial time while it is NP-hard for general chordal graphs.

Twist polynomials of delta-matroids

Recently, Gross, Mansour and Tucker introduced the partial duality polynomial of a ribbon graph and posed a conjecture that there is no orientable ribbon graph whose partial duality polynomial has only one non-constant term. We found an infinite family of counterexamples for the conjecture and showed that essentially these are the only counterexamples. This is also obtained independently by Chumutov and Vignes-Tourneret and they posed a problem: it would be interesting to know whether the partial duality polynomial and the related conjectures would make sense for general delta-matroids. In this paper, we show that partial duality polynomials have delta-matroid analogues. We introduce the twist polynomials of delta-matroids and discuss its basic properties for delta-matroids. We give a characterization of even normal binary delta-matroids whose twist polynomials have only one term and then prove that the twist polynomial of a normal binary delta-matroid contains non-zero constant term if and only if its intersection graph is bipartite.

Lucky \(k\)-polynomials of null and complete split graphs

The concept of Lucky colorings of a graph is used to introduce the notion of the Lucky \(k\)-polynomials of null graphs. We then give the Lucky \(k\)-polynomials for complete split graphs and generalized star graphs. Finally, further problems of research related to this concept are discussed.

On the location of zeros of the Laplacian matching polynomials of graphs

The Laplacian matching polynomial of a graph G, denoted by \(\mathscr {LM}(G,x)\), is a new graph polynomial whose all zeros are nonnegative real numbers. In this paper, we investigate the location of zeros of the Laplacian matching polynomials. Let G be a connected graph. We show that 0 is a zero of \(\mathscr {LM}(G, x)\) if and only if G is a tree. We prove that the number of distinct positive zeros of \(\mathscr {LM}(G,x)\) is at least equal to the length of the longest path in G. It is also established that the zeros of \(\mathscr {LM}(G,x)\) and \(\mathscr {LM}(G-e,x)\) interlace for each edge e of G. Using the path tree of G, we present a linear algebraic approach to investigate the largest zero of \(\mathscr {LM}(G,x)\) and particularly to give tight upper and lower bounds on it.