Polynomial Of Graph Research Articles

Base polynomials for degree and distance based topological invariants of n-bilinear straight pentachain

Degree and distance-based graph polynomials are important not only as graph invariants but also for their applications in physics, chemistry, and pharmacy. The present paper is concerned with the Hosoya and Schultz polynomials of n-bilinear straight pentachain. It was observed that computing these polynomials directly by definitions is extremely difficult. Thus, we follow the divide and conquer rule and introduce the idea of base polynomials. We actually split the vertices into disjoint classes and then wrote the number of paths in terms of polynomials for each class, which ultimately serve as bases for Hosoya and Schultz polynomials. We also recover some indices from these polynomials. Finally, we give an example to show how these polynomials actually work.

Laplacian spectra of power graphs of certain prime-power Abelian groups

The power graph [Formula: see text] of a semigroup [Formula: see text] is a simple undirected graph whose vertex set is [Formula: see text] itself, and any two distinct vertices are adjacent if one of them is a power of the other. In this paper, we describe the power graph [Formula: see text] in terms of joins and disjoint unions of complete graphs, and use this to calculate the Laplacian polynomial of [Formula: see text]. We finally calculate the Laplacian polynomial of the power graph [Formula: see text].

Partial duality for ribbon graphs, III: a Gray code algorithm for enumeration

Partially Poincare-dualizing an embedded graph G on an arbitrary subset of edges was defined geometrically by Chmutov, using ribbon graphs. Part I of this series of papers introduced the partial-duality polynomial, which enumerates all the possible partial duals of the graph G, according to their Euler-genus, which can change according to the selection of the edge subset on which to dualize. Ellis-Monaghan and Moffatt have expanded the partial-duality concept to include the Petrie dual, the Wilson dual, and the two triality operators. Abrams and Ellis-Monaghan have given the five operators the collective name twualities. Part II of this series of papers derived formulas for partial-twuality polynomials corresponding to several fundamental sequences of embedded graphs. Here in Part III, we present an algorithm to calculate the partial-twuality polynomial of a ribbon graph G, for all twualities, which involves organizing the edge subsets of G into a hypercube and traversing that hypercube via a Gray code.

Upper Bounds for the Independence Polynomial of Graphs at -1

The independence polynomial of a graph G is $$I(G;x)=\sum _{k=0}^{\alpha (G)} s_{k}\cdot x^{k}$$ , where $$s_{k}$$ and $$\alpha (G)$$ denote the number of independent sets of cardinality k and the independence number of G, respectively. We say that a cycle is a $$\tilde{3}$$ -cycle if its length is divisible by 3, otherwise a non- $$\tilde{3}$$ -cycle. Define $$\phi (G)$$ to be the decycling number of G. Engstrom proved that $$|I(G;-1)| \le 2^{\phi (G)}$$ for any graph G. In this paper, we first prove that $$|I(G;-1)|\le 2^{\beta (G)}-\beta (G)$$ for graphs with non- $$\tilde{3}$$ -cycles, where $$\beta (G)$$ is the cyclomatic number of G. Infinitely many examples show that there do exist graphs satisfying $$\beta (G)=\phi (G)$$ and containing non- $$\tilde{3}$$ -cycles. In such a case, it improves the Engstrom’s result. Furthermore, $$|I(G;-1)|\le 2^{k}$$ provided that all cycles of G are pairwise disjoint, where k is the number of $$\tilde{3}$$ -cycles of G. This provides a new perspective on estimating the independence polynomial at -1 of many special graphs. In the case G contains no vertices of degree 1, $$|I(G;-1)|\le 2^{\beta (G)}-1$$ if $$\beta (G)\ge 2$$ , and $$|I(G;-1)|\le 2^{\beta (G)-1}$$ if G contains a non- $$\tilde{3}$$ -cycle.

Strict log-concavity of the Kirchhoff polynomial and its applications to the strong Lefschetz property

Anari, Gharan, and Vinzant proved (complete) log-concavity of the basis generating functions for all matroids. From the viewpoint of combinatorial Hodge theory, it is natural to ask whether these functions are “strictly” log-concave for simple matroids. In this paper, we show this strictness for simple graphic matroids, that is, we show that Kirchhoff polynomials of simple graphs are strictly log-concave. Our key observation is that the Kirchhoff polynomial of a complete graph can be seen as the (irreducible) relative invariant of a certain prehomogeneous vector space, which may be independently interesting in its own right. Furthermore, we prove that for any ai∈R>0, a1x1+⋯+anxn∈RM1 satisfies the strong Lefschetz property (moreover, Hodge–Riemann bilinear relation) at degree one of the Artinian Gorenstein algebra RM⁎ associated to a graphic matroid M, which is defined by Maeno and Numata for all matroids.

Mixed partition functions and exponentially bounded edge-connection rank

We study graph parameters whose associated edge-connection matrices have exponentially bounded rank growth. Our main result is an explicit construction of a large class of graph parameters with this property that we call mixed partition functions . Mixed partition functions can be seen as a generalization of partition functions of vertex models, as introduced by de la Harpe and Jones, [P. de la Harpe and V. F. R. Jones, Graph invariants related to statistical mechanical models: examples and problems, J. Combin. Theory Ser. B 57 (1993), no. 2, 207–227.] and they are related to invariant theory of orthosymplectic supergroup. We moreover show that evaluations of the characteristic polynomial of a simple graph are examples of mixed partition functions, answering a question of de la Harpe and Jones. (NOTE. Some of the results of this paper were announced in an extended abstract: G. Regts and B. Sevenster, Partition functions from orthogonal and symplectic group invariants, Electron. Notes Discrete Math. 61 (2017), 1011–1017. Unfortunately that reference contains a mistake; we will comment on that below).

Anomalous IoT Sensor Data Detection: An Efficient Approach Enabled by Nonlinear Frequency-Domain Graph Analysis

The detection of anomalous Internet-of-Things (IoT) sensor data is extremely important in many industrial applications due to the catastrophic consequences of the faulty or unreliable sensor data. Good anomalous data detection performance with high detection efficiency is indeed a dilemma since it is difficult to derive an explicit detection function to characterize the relationships between the anomalous values and the detection indicator. To overcome this difficulty, the location information of the IoT sensors is exploited in this study to characterize and reconstruct the relationships among the sensor data based on a second-order nonlinear polynomial graph filter (NPGF). The analysis of the sensor data reconstruction model is then conducted in the frequency domain based on the 2-D inverse graph Fourier transform (GFT), and the reconstruction error function for the sensor data is analytically derived based on the second-order GFT coefficients. It is shown that the detection efficiency can be greatly improved if the input graph signal is designed to be bandlimited. The anomalous sensor data detection is then conducted in the frequency domain as high-frequency components are more sensitive to the deviation values. An NPGF-based frequency-domain algorithm is proposed for the anomalous sensor data detection, which is illustrated and validated with a real-world data set for temperature monitoring. The simulation results demonstrate the detection performance and efficiency improvement of the proposed algorithm in anomaly detection.

Some Topological Indices and ( Hosoya and Schultz) Polynomial of Subgroup intersection graph of a group Z_r

Let be any group with identity element (e) . A subgroup intersection graph of a group is the Graph with V (G) = - e and two separate a vertices a and b contiguous for a and b if and only if , Where is a cyclic subgroup of generated by . We found some topological indices in this paper and (Hosoya and Schultz) Polynomial of , where , is a prime number.

Harary polynomials

Given a graph property $\mathcal{P}$, F. Harary introduced in 1985 $\mathcal{P}$-colorings, graph colorings where each colorclass induces a graph in $\mathcal{P}$. Let $\chi_{\mathcal{P}}(G;k)$ counts the number of $\mathcal{P}$-colorings of $G$ with at most $k$ colors. It turns out that $\chi_{\mathcal{P}}(G;k)$ is a polynomial in $\mathbb{Z}[k]$ for each graph $G$. Graph polynomials of this form are called Harary polynomials. In this paper we investigate properties of Harary polynomials and compare them with properties of the classical chromatic polynomial $\chi(G;k)$. We show that the characteristic and Laplacian polynomial, the matching, the independence and the domination polynomial are not Harary polynomials. We show that for various notions of sparse, non-trivial properties $\mathcal{P}$, the polynomial $\chi_{\mathcal{P}}(G;k)$ is, in contrast to $\chi(G;k)$, not a chromatic, and even not an edge elimination invariant. Finally we study whether Harary polynomials are definable in Monadic Second Order Logic.

Fractional Order Graph Filters: Design and Implementation

Existing graph filters, polynomial or rational, are mainly of integer order forms. However, there are some frequency responses which are not easily achieved by integer order approximation. It will substantially increase the flexibility of the filters if we relax the integer order to fractional ones. Motivated by fractional order models, we introduce the fractional order graph filters (FOGF), and propose to design the filter coefficients by genetic algorithm. In order to implement distributed computation on a graph, an FOGF can be approximated by the continued fraction expansion and transformed to an infinite impulse response graph filter.

Euler-Genus Distributions of Cubic Caterpillar-Halin Graphs

In 2011, Gross derived an $$O(n^2)$$ -time algorithm to calculate the genus distribution of a given cubic Halin graph. In this paper, with the help of the overlap matrix, we obtain a recurrence relation for the Euler-genus polynomials of cubic caterpillar-Halin graphs. Explicit formulas for the embeddings of the cubic caterpillar-Halin graphs into surfaces with Euler-genus 0, 1 and 2 are also obtained.

On a poset of trees revisited

This contribution gives an extensive study on the Wiener indices, the number of closed walks, the coefficients of some graph polynomials (the adjacency polynomial, the Laplacian polynomial, the edge cover polynomial and the independence polynomial) of trees. Csikvári (2010) [4] introduced the generalized tree shift, which keeps the number of vertices of trees. Applying the generalized tree shifts and recurrence relation, we extend the works of Csikvári (2010) [4] and Csikvári (2013) [5]. Using a unified approach, we obtain the following main results: Firstly, for all n and ℓ, we characterize the unique tree having the maximum (resp. minimum) Wiener index and the unique tree having the maximum (resp. minimum) number of closed walks of length ℓ among the trees of order n which are neither a path nor a star. Secondly, we characterize the unique tree whose adjacency polynomial (resp. Laplacian polynomial, independence polynomial) has the maximum (resp. minimum) coefficients in absolute value among the trees of order n which are neither a path nor a star, respectively. At last, we identify all the n-vertex trees whose edge cover polynomial has the minimum coefficients in absolute value and we also determine the unique tree having the maximum coefficients in absolute value among the trees of order n which are neither a path nor a star.

The restricted detour polynomials of a prism and some wheel related graphs

The restricted detour index of a connected graph G is equal to the sum of restricted detour distances of all unordered pairs of vertices of G. The restricted detour polynomial of graph G is defined as , where C*(G, k) is the number of unordered pairs of vertices of G laying at restricted detour distance k from each other and δ * is restricted detour diameter of G. In this paper, restricted detour polynomials and restricted detour indices of prism graphs, helm graphs and closed helm graphs have been computed.

Nitrogen Metabolism and Microbial Protein Synthesis by Local Sheep Fed Diet Containing Hibiscus Leave Meal (HLM) with Different Direct-Fed Microbials (DFM) Supplementation

The interaction of Hibiscus Leaf Meal (HLM) and Direct-Fed Microbials (DFM) supplementation on nitrogen metabolism and rumen microbial synthesis by local sheep was investigated. Thirty six male local sheep aged ±2 years, weight 28.01±2.61 kg were fed concentrate (3% of body weight) supplemented with HLM twice a day, amoniated rice-straw (ARS) supplemented with DFM were given ad-libitum were assigned randomly to nine treatment in an experiment of 3×3 factorial design. The first factor (P) was DFM were supplemented in ARS (P0=without DFM, P1=DFMAMS, and P2=DFMRK). Second factor (W) was the level of HLM supplementation in concentrate (W0=0%, W1=0.24% and W2=0.48% of DM consentrate. The study measured variables including nitrogen digestibility (ND), nitrogen retention (NR), microbial protein synthesis (MPS), and Efficiency of microbial protein synthesis (EMPS). There were significant interaction between DFM and HLM suplementation on MPS and EMPS. However, the interaction of ND and NR was non-existent. Based on polynomial graph, the most efficient MPS was achieved on combination between DFMRK and 0.23% HLM about 8.70 gN/day. Despite the absence of interaction (P>0.05) between DFM supplementation and HLM on KN and RN, DFM supplementation significantly affected (P<0.01) Nitrogen metabolism (ND and NR). The treatment without DFM supplementation resulted the highest nitrogen metabolism (ND and NR). This study concludes that HLM without DFM suplementation positively impacted N Metabolism. Combination between 0.23% HLM and DFMRK resulted the most efficient Microbial Protein Synthesis.

Numerical Solution for Nonlinear Klein–Gordon Equation via Operational Matrix by Clique Polynomial of Complete Graphs

This study introduced a generalized operational matrix using Clique polynomials of a complete graph and proposed the latest approach to solve the non-linear Klein–Gordon (KG) equation. KG equations describe many real physical phenomena in fluid dynamics, electrical engineering, biogenetics, tribology. By using the properties of the operational-matrix, we transform-the non-linear KG equation into a system-of algebraic-equations. Unknown coefficients to be determined by Newton’s method. The present-technique is applied-to four problems, and the obtained-results are-compared with-another-method in the literature. Also, we discussed some theorems on convergence analysis and continuous property.

The Independent Set Polynomials of a class of Graphs

In this paper, a special graph is defined. According to the structural characteristics of the special graph , the independent set polynomial of a special graph is derived by using the properties and lemma of independent set polynomial of simple graph. The Merrifield-Simmons index of this kind of special graph can be obtained.

M-polynomial-based topological indices of metal-organic networks

Abstract Topological index (TI) is a numerical invariant that helps to understand the natural relationship of the physicochemical properties of a compound in its primary structure. George Polya introduced the idea of counting polynomials in chemical graph theory and Winer made the use of TI in chemical compounds working on the paraffin's boiling point. The literature of the topological indices and counting polynomials of different graphs has grown extremely since that time. Metal-organic network (MON) is a group of different chemical compounds that consist of metal ions and organic ligands to represent unique morphology, excellent chemical stability, large pore volume, and very high surface area. Working on structures, characteristics, and synthesis of various MONs show the importance of these networks with useful applications, such as sensing of different gases, assessment of chemicals, environmental hazard, heterogeneous catalysis, gas and energy storage devices of excellent material, conducting solids, super-capacitors and catalysis for the purification, and separation of different gases. The above-mentioned properties and physical stability of these MONs become a most discussed topic nowadays. In this paper, we calculate the M-polynomials and various TIs based on these polynomials for two different MONs. A comparison among the aforesaid topological indices is also included to represent the better one.

On intersection graph of dihedral group

Let $G$ be a finite group. The intersection graph of $G$ is a graph whose vertex set is the set of all proper non-trivial subgroups of $G$ and two distinct vertices $H$ and $K$ are adjacent if and only if $H\cap K \neq \{e\}$, where $e$ is the identity of the group $G$. In this paper, we investigate some properties and exploring some topological indices such as Wiener, Hyper-Wiener, first and second Zagreb, Schultz, Gutman and eccentric connectivity indices of the intersection graph of $D_{2n}$ for $n=p^2$, $p$ is prime. We also find the metric dimension and the resolving polynomial of the intersection graph of $D_{2p^2}$.

Equi Neighbor Polynomial of Graphs

Equi Neighbor Polynomial of Graphs

Distributed Nonlinear Polynomial Graph Filter and Its Output Graph Spectrum: Filter Analysis and Design

While frequency-domain algorithms have been demonstrated to be powerful for conventional nonlinear signal processing, there is still not much progress in literature dedicated to nonlinear graph signal processing in frequency domain. The difficulties come from the irregular structure of graph signals and the lack of a straightforward description for the nonlinear relationship between input and output graph signals. In this paper, a distributed nonlinear polynomial graph filter (NPGF) is proposed to characterize the nonlinear input-output relationship by employing polynomial-type nonlinearity based on Hadamard product. Then, the nonlinear output graph signal is reformulated based on the multi-dimensional inverse graph Fourier transform (GFT), where the m-dimensional GFT coefficients are defined as explicit functions of input graph spectrum and filter structural factors (filter parameters, graph shift matrix, nonlinear degree, and graph shift order). The effects of filter structural factors on the output graph spectrum are further evaluated by proposing graph frequency response (GFR) coefficients, which inherently characterize the irregular structural behaviors of a distributed NPGF in frequency domain. The nonlinear output graph spectrum is then developed based on GFR vectors by mapping the GFR coefficients in the tensor formulation to the output GFT coefficients in the vector expression. This can greatly improve the efficiency in calculating the output graph signal if the input signal only involves a few frequency components. Finally, algorithms are proposed to design the filter parameters for the desired output signal. Numerical studies are presented to illustrate and validate the proposed frequency-domain framework for nonlinear graph signal processing.