PROPERTIES OF SPECTRA OF GRAPHS AND LINE GRAPHS

Let G be a simple connected graph with n vertices and m edges,L G be the line graph of G and λ 1(L G)≥λ 2(L G)≥...≥λ m(L G) be the eigenvalues of the graph L G.In this paper,the range of eigenvalues of a line graph is considered.Some sharp upper bounds and sharp lower bounds of the eigenvalues of L G are obtained.In particular,it is proved that-2cos(πn)≤λ n-1 (L G)≤n-4 and λ n(L G)=-2 if and only if G is bipartite.

P_3-factor in Line Graphs of Trees

In this paper,we give a necessary and sufficient condition for the existence of P3-factors in the line graph of a tree.Then we present an algorithm to determine whether the line graph of a tree has a P3-factor.

PANCYCLICITY IN LINE GRAPHS

This paper shows that if G is a connected graph of order n such that <(sigma(2)(G))over bar> > 2(n/5 - 1) and L(G) is hamiltonian, then, for n greater than or equal to 43, L(G) is pancyclic. Using the result of Veldman([8]) this result settles the conjecture of Benhocine, et.al([1]): Let G be a connected almost bridgeless graph of order n such that <(sigma(2)(G))over bar> > 2(n/5 - 1). If n is sufficintly large, L(G) is pancyclic.

On traceable iterated line graph and hamiltonian path index

Xiong and Liu[21]gave a characterization of the graphs G for which the n-iterated line graph L^(n)(G)is hamiltonian,for n≥2.In this paper,we study the existence of a hamiltonian path in L^(n)(G),and give a characterization of G for which L^(n)(G)has a hamiltonian path.As applications,we use this characterization to give several upper bounds on the hamiltonian path index of a graph.

Super Connectivity of Line Graphs and Digraphs

The h-super connectivity κh and the h-super edge-connectivity λh are more refined network reliability indices than the conneetivity and the edge-connectivity. This paper shows that for a connected balanced digraph D and its line digraph L, if D is optimally super edge-connected, then κ1（L） = 2λ1 （D）, and that for a connected graph G and its line graph L, if one of κ1 （L） and λ（G） exists, then κ1（L） = λ2（G）. This paper determines that κ1（B（d, n） is equal to 4d- 8 for n = 2 and d ≥ 4, and to 4d-4 for n ≥ 3 and d ≥ 3, and that κ1（K（d, n）） is equal to 4d- 4 for d 〉 2 and n ≥ 2 except K（2, 2）. It then follows that B（d,n） and K（d, n） are both super connected for any d ≥ 2 and n ≥ 1.

Multigraphic Degree Sequences and Hamiltonian-connected Line Graphs

Let G be a multigraph.Suppose that e=u1v1 and e′=u2v2 are two edges of G.If e≠e′,then G(e,e′)is the graph obtained from G by replacing e=u1v1 with a path u1vev1 and by replacing e′=u2v2 with a path u2ve′v2,where ve,ve′are two new vertices not in V(G).If e=e′,then G(e,e′),also denoted by G(e),is obtained from G by replacing e=u1v1 with a path u1vev1.A graph G is strongly spanning trailable if for any e,e′∈E(G),G(e,e′)has a spanning(ve,ve′)-trail.The design of n processor network with given number of connections from each processor and with a desirable strength of the network can be modelled as a degree sequence realization problem with certain desirable graphical properties.A sequence d=(d1,d2,⋯,dn)is multigraphic if there is a multigraph G with degree sequence d,and such a graph G is called a realization of d.A multigraphic degree sequence d is strongly spanning trailable if d has a realization G which is a strongly spanning trailable graph,and d is line-hamiltonian-connected if d has a realization G such that the line graph of G is hamiltonian-connected.In this paper,we prove that a nonincreasing multigraphic sequence d=(d1,d2)⋯,dn)is strongly spanning trailable if and only if either n=1 and d1=0 or n≥2 and dn≥3.Applying this result,we prove that for a nonincreasing multigraphic sequence d=(d1,d2,⋯,dn),if n≥2 and dn≥3,then d is line-hamiltonian-connected.

Wiener Index of Graphs and Their Line Graphs

The Wiener index W(G)of a graph G is a distance-based topological index defined as the sum of distances between all pairs of vertices in G.It is shown that forλ=2 there is an infinite family of planar bipartite chemical graphs G of girth 4 with the cyclomatic numberλ,but their line graphs are not chemical graphs,and forλ≥2 there are two infinite families of planar nonbipartite graphs G of girth 3 with the cyclomatic numberλ;the three classes of graphs have the property W(G)=W(L(G)),where L(G)is the line graph of G.

Even factors with a bounded number of components in iterated line graphs

We consider even factors with a bounded number of components in the n-times iterated line graphs L^n(G). We present a characterization of a simple graph G such that L^n(G) has an even factor with at most k components, based on the existence of a certain type of subgraphs in G. Moreover, we use this result to give some upper bounds for the minimum number of components of even factors in L^n(G) and also show that the minimum number of components of even factors in L^n(G) is stable under the closure operation on a claw-free graph G, which extends some known results. Our results show that it seems to be NP-hard to determine the minimum number of components of even factors of iterated line graphs. We also propose some problems for further research.

Nomogram model including LATS2 expression was constructed to predict the prognosis of advanced gastric cancer after surgery

BACKGROUND Gastric cancer is a leading cause of cancer-related deaths worldwide.Prognostic assessments are typically based on the tumor-node-metastasis(TNM)staging system,which does not account for the molecular heterogeneity of this disease.LATS2,a tumor suppressor gene involved in the Hippo signaling pathway,has been identified as a potential prognostic biomarker in gastric cancer.AIM To construct and validate a nomogram model that includes LATS2 expression to predict the survival prognosis of advanced gastric cancer patients following ra-dical surgery,and compare its predictive performance with traditional TNM staging.METHODS A retrospective analysis of 245 advanced gastric cancer patients from the Fourth Hospital of Hebei Medical University was conducted.The patients were divided into a training group(171 patients)and a validation group(74 patients)to deve-lop and test our prognostic model.The performance of the model was determined using C-indices,receiver operating characteristic curves,calibration plots,and decision curves.RESULTS The model demonstrated a high predictive accuracy with C-indices of 0.829 in the training set and 0.862 in the validation set.Area under the curve values for three-year and five-year survival prediction were significantly robust,suggesting an excellent discrimination ability.Calibration plots confirmed the high concordance between the predictions and actual survival outcomes.CONCLUSION We developed a nomogram model incorporating LATS2 expression,which significantly outperformed conven-tional TNM staging in predicting the prognosis of advanced gastric cancer patients postsurgery.This model may serve as a valuable tool for individualized patient management,allowing for more accurate stratification and im-proved clinical outcomes.Further validation in larger patient cohorts will be necessary to establish its generaliza-bility and clinical utility.

On λ-harmonic graphs

A λ harmonic graph G, a λ-Hgraph G for short, means that there exists a constant A such that the equality λd（ui） = ∑（vi,vj）∈E（G） d（vj） holds for all i = 1, 2,…, ｜V（G）｜, where d（vi） denotes the degree of vertex vi. In this paper, some harmonic properties of the complement and line graph are given, and some algebraic properties for the λ-Hgraphs are obtained.

Interpretation of data and treatment of Jing-well point temperatures test

The Jing-well point temperatures test method is a method to diagnose and guide the treatment of diseases by measuring the subjects' symmetrical well point temperature. it is improved from the method of knowing heat sensitivity. The application of Jing-well point temperatures test method is wide, and it can be used in internal and external gynecology and pediatrics and facial features department. at the same time, it has the advantage of objective and accurate diagnosis. The old law has some shortcomings, such as poor intuition, unavoidable omission of information, incomplete interpretation of information and so on. In this paper, Excel software is used to transform the data into line chart form, which improves the intuition and comprehensiveness of this method, so that the data can be better interpreted and used. It is newly proposed in this article that in addition to observing the longitudinal di fference of well point temperature, more attention should be paid to the horizontal contrast difference of well point temperature in different meridians. The article also summarizes a number of treatment methods, including acupuncture, moxa moxibustion, cupping and scraping, and the selection of acupoints, including mother acupoints, tenderness points and heat-sensitive moxibustion, so that doctors can combine traditional Chinese medicine professional knowledge in clinic.

Degree sum conditions for hamiltonian index

In this note,we show a sharp lower bound of min{Σ_(i=1)^(k)dG(u_(i)):u1u2...uk is a path of(2-)connected G}on its order such that(k-1)-iterated line graphs L^(k-1)(G)are hamiltonian.

L(1, 2)-edge-labelings for lattices

For a graph G and two positive integers j and k, an m-L（j, k）-edge-labeling of G is an assignment on the edges to the set {0, 1, 2,..., m}, such that adjacent edges which receive labels differ at least by j, and edges which are distance two apart receive labels differ at least by k. The λj,k-number of G is the minimum m such that an m-L（j, k）-edge-labeling is admitted by G. In this article, the L（1, 2）-edge-labeling for the hexagonal lattice, the square lattice and the triangular lattice are studied, and the bounds for λj,k-numbers of these graphs are obtained.

On the Supereulerian Index of a Graph

Two methods for determining the supereulerian index of a graph G are given. A sharp upper bound and a sharp lower bound on the supereulerian index by studying the branch bonds of G are got.

The Wiener Index of an Undirected Power Graph

The undirected power graph P(Zn) of a finite group Zn is the graph with vertex set G and two distinct vertices u and v are adjacent if and only if u ≠ v and or . The Wiener index W(P(Zn)) of an undirected power graph P(Zn) is defined to be sum of distances between all unordered pair of vertices in P(Zn). Similarly, the edge-Wiener index We(P(Zn)) of P(Zn) is defined to be the sum of distances between all unordered pairs of edges in P(Zn). In this paper, we concentrate on the wiener index of a power graph , P(Zpq) and P(Zp). Firstly, we obtain new results on the wiener index and edge-wiener index of power graph P(Zn), using m,n and Euler function. Also, we obtain an equivalence between the edge-wiener index and wiener index of a power graph of Zn.

Line Graph Links

It is well known that a shaded link diagram corresponds to a signed plane multi-graph.In graph theory,line graph is an old and important concept originally introduced by H.Whitney in 1932.In this paper we define the line graph link to be a link which has a diagram whose corresponding signed plane graph is a signed line graph.The main purpose of the paper is to illustrate the structure of planar line graphs,which permits us to deal with its signed Tutte polynomial and the Jones polynomials of line graph links.

Clique-Transversal Sets in 4-Regular Claw-Free Graphs

A clique-transversal set D of a graph C is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by To（G）, is the minimum cardinality of a clique- transversal set in G. In this paper we give the exact value of the clique-transversal number for the line graph of a complete graph. Also, we give a lower bound on the clique-transversal number for 4-regular claw-free graphs and characterize the extremal graphs achieving the lower bound.

Bounds for the Rainbow Disconnection Numbers of Graphs

An edge-cut of an edge-colored connected graph is called a rainbow cut if no two edges in the edge-cut are colored the same.An edge-colored graph is rainbow disconnected if for any two distinct vertices u and v of the graph,there exists a rainbow cut separating u and v.For a connected graph G,the rainbow disconnection number of G,denoted by rd(G),is defined as the smallest number of colors required to make G rainbow disconnected.In this paper,we first give some upper bounds for rd(G),and moreover,we completely characterize the graphs which meet the upper bounds of the NordhausGaddum type result obtained early by us.Secondly,we propose a conjecture that for any connected graph G,either rd(G)=λ^(+)(G)or rd(G)=λ^(+)(G)+1,whereλ^(+)(G)is the upper edge-connectivity,and prove that the conjecture holds for many classes of graphs,which supports this conjecture.Moreover,we prove that for an odd integer k,if G is a k-edge-connected k-regular graph,thenχ’(G)=k if and only if rd(G)=k.It implies that there are infinitely many k-edge-connected k-regular graphs G for which rd(G)=λ^(+)(G)for odd k,and also there are infinitely many k-edge-connected k-regular graphs G for which rd(G)=λ^(+)(G)+1 for odd k.For k=3,the result gives rise to an interesting result,which is equivalent to the famous Four-Color Problem.Finally,we give the relationship between rd(G)of a graph G and the rainbow vertex-disconnection number rvd(L(G))of the line graph L(G)of G.

Interpreting the vulnerability of power systems in cascading failures using multi-graph convolutional networks

Analyzing the vulnerability of power systems in cascading failures is generally regarded as a challenging problem. Although existing studies can extract some critical rules, they fail to capture the complex subtleties under different operational conditions. In recent years, several deep learning methods have been applied to address this issue. However, most of the existing deep learning methods consider only the grid topology of a power system in terms of topological connections, but do not encompass a power system’s spatial information such as the electrical distance to increase the accuracy in the process of graph convolution. In this paper, we construct a novel power-weighted line graph that uses power system topology and spatial information to optimize the edge weight assignment of the line graph. Then we propose a multi-graph convolutional network(MGCN) based on a graph classification task, which preserves a power system’s spatial correlations and captures the relationships among physical components. Our model can better handle the problem with power systems that have parallel lines, where our method can maintain desirable accuracy in modeling systems with these extra topology features. To increase the interpretability of the model, we present the MGCN using layer-wise relevance propagation and quantify the contributing factors of model classification.