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### Edge-neighbor-integrity of trees

Margaret B. Cozzens, Shu-Shih Y. Wu
1994 The Australasian Journal of Combinatorics
In this paper, we find the minimum and maximum edge-neighbor-integrity among all trees with any fixed order, and also show that for any integer I between the extreme values there is a tree with the edge-neighbor-integrity  ...  The edge-neighbor-integrity of a graph G, ENI(G), is defined to be ENI(G) = min {lSI + w(G/S)}, where S is any edge subversion strategy of G, and w(G/S) is the maximum order of the components of GIS.  ...  If we can find a tree of order at least 3 whose edge-neighbor-integrity is 2, then the minimum edge-neighbor-integrity among all trees is 2. QED.  ...

### Discrete SIR model on a homogeneous tree and its continuous limit [article]

2022 arXiv   pre-print
We study a discrete Susceptible-Infected-Recovered (SIR) model for the spread of infectious disease on a homogeneous tree and the limit behavior of the model in the case when the tree vertex degree tends  ...  We obtain the distribution of the time it takes for a susceptible vertex to get infected in terms of a solution of a non-linear integral equation under broad assumptions on the model parameters.  ...  Further, recall that a rooted homogeneous tree with the vertex degree N ≥ 2 as a tree, where one of the vertices, called the root, has N − 1 neighbors, while any other vertex has N neighbors.  ...

### Vertex-neighbor-integrity of magnifiers, expanders, and hypercubes

Marci J. Gambrell
2000 Discrete Mathematics
The vertex-neighbor-integrity of G is deÿned to be VNI is the order of the largest connected component in the graph H .  ...  Finally, we show that the decision problem corresponding to computing the vertex-neighbor-integrity of a graph is NP-complete.  ...  This research was done while the author was an undergraduate at the University of Chicago under the supervision of Joseph Gallian at the 1998 Summer Research Program at the University of Minnesota, Duluth  ...

### Integral trees with given nullity [article]

E. Ghorbani, A. Mohammadian, B. Tayfeh-Rezaie
2015 arXiv   pre-print
A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees.  ...  It is also shown that integral trees with nullity 2 and 3 are unique.  ...  Acknowledgments The authors are indebted to an anonymous referee for pointing out an error in the proof of the main result in an earlier version of the manuscript.  ...

### A linear delay algorithm for enumerating all connected induced subgraphs

Mohammed Alokshiya, Saeed Salem, Fidaa Abed
2019 BMC Bioinformatics
To efficiently mine all maximal cohesive subgraphs, we propose two pruning techniques that remove futile search nodes in the enumeration tree.  ...  Building on this enumeration approach, we propose an algorithm for mining all maximal cohesive subgraphs that integrates vertices' attributes with subgraph enumeration.  ...  Availability of data and materials The dataset and the implementation of the algorithm is available at http:// www.cs.ndsu.nodak.edu/~ssalem/multirelation.html. About this supplement  ...

### The integrally representable trees of norm 3 [article]

Jack H. Koolen, Masood Ur Rehman, Qianqian Yang
2016 arXiv   pre-print
In this paper, we determine the integrally representable trees of norm 3.  ...  (ii) If h contains a slim vertex with at least two fat neighbors, then h is integrally representable of norm 3. 3 Tree-like Hoffman graphs Stripped Hoffman graphs Let t 1 and t 2 be two tree-like Hoffman  ...  For a vertex x of h, we define N s h (x) (resp. N f h (x) ) the set of slim (resp. fat) neighbors of x in h. If every slim vertex of the Hoffman graph h has a fat neighbor, then we call h fat.  ...

### The integrally representable trees of norm \$3\$

Jack H. Koolen, Masood Ur Rehman, Qianqian Yang
2017 Annals of Mathematical Sciences and Applications
For a 3-seedling t with ψ as its reduced representation of norm 3, let Λ red (t, 3) be the lattice generated by vectors ψ(x) with x ∈ V s (h). Problem 2.  ...  Recall from  that an integral lattice Λ is s-integrable, for a positive integer s, if it can be described by vectors s − 1 2 (x 1 , . . . , x n ) with all x i ∈ Z.  ...  (ii) If h contains a slim vertex with at least two fat neighbors, then h is integrally representable of norm 3. Lemma 3 . 4 . 34 Let T = (T, sgn) be an edge-signed tree.  ...

### Integral trees with given nullity

E. Ghorbani, A. Mohammadian, B. Tayfeh-Rezaie
2016 Discrete Mathematics
A graph is called integral if all eigenvalues of its adjacency matrix consist entirely of integers. We prove that for a given nullity more than 1, there are only finitely many integral trees.  ...  Integral trees with nullity at most 1 were already characterized by Watanabe and Brouwer. It is shown that integral trees with nullity 2 and 3 are unique.  ...  Acknowledgments The authors are indebted to an anonymous referee for pointing out an error in the proof of the main result in an earlier version of the manuscript.  ...

### Statistics of planar graphs viewed from a vertex: a study via labeled trees

J. Bouttier, P. Di Francesco, E. Guitter
2003 Nuclear Physics B
We study the statistics of edges and vertices in the vicinity of a reference vertex (origin) within random planar quadrangulations and Eulerian triangulations.  ...  In the case of infinitely large graphs, we give in particular explicit formulas for the probabilities that the origin have given numbers of neighboring edges and/or vertices, as well as explicit values  ...  For each vertex of the tree reaching position 1, we have an energy gain ρα k , where k is the number of nearest neighbors of the vertex on the tree.  ...

### Bijective Deformations in R^n via Integral Curve Coordinates [article]

Lisa Huynh, Yotam Gingold
2015 arXiv   pre-print
We introduce Integral Curve Coordinates, which identify each point in a bounded domain with a parameter along an integral curve of the gradient of a function f on that domain; suitable functions have exactly  ...  Because every integral curve intersects the boundary exactly once, Integral Curve Coordinates provide a natural bijective mapping from one domain to another given a bijection of the boundary.  ...  Cousin tree vertex relationships: Two vertices u, v that are axis-aligned neighbors in the grid must be either parent-child (left) or tree cousins (right).  ...

### Implementation Of Heuristics For Solving Travelling Salesman Problem Using Nearest Neighbour And Minimum Spanning Tree Algorithms

Fatma A. Karkory, Ali A. Abudalmola
2013 Zenodo
This paper adopts the nearest neighbor and minimum spanning tree algorithm to solve the well-known travelling salesman problem. The algorithms were implemented using java programming language.  ...  The approach is tested on three graphs that making a TSP tour instance of 5-city, 10 –city, and 229–city. The computation results validate the performance of the proposed algorithm.  ...  The root of the tree is the start vertex or point in the tour. First we iterate over the root's neighbors; for each neighbor w, we add the edge triple <root, w, weight> to the priority queue.  ...

### RAG: RNA-As-Graphs web resource

Daniela Fera, Namhee Kim, Nahum Shiffeldrim, Julie Zorn, Uri Laserson, Hin Hark Gan, Tamar Schlick
2004 BMC Bioinformatics
All RNA motifs are filed according to their graph vertex number (RNA length) and ranked by topological complexity.  ...  The proliferation of structural and functional studies of RNA has revealed an increasing range of RNA's structural repertoire.  ...  Fera acknowledges support from the Dean's Undergraduate Research Fund and a summer fellowship from the Department of Chemistry.  ...

### Walks and regular integral graphs

Dragan Stevanović, Nair M.M. de Abreu, Maria A.A. de Freitas, Renata Del-Vecchio
2007 Linear Algebra and its Applications
in an infinite regular tree, provides a lower bound on the number of closed walks in regular graphs.  ...  This lower bound is then applied to reduce the number of the feasible spectra of the 4-regular bipartite integral graphs by more than a half.  ...  It is homomorphic by its construction, since the neighbors of any vertex u ∈ V (B r ) are mapped to the neighbors of f (u) in G.  ...

### Acquisition-extremal graphs

Timothy D. LeSaulnier, Douglas B. West
2013 Discrete Applied Mathematics
three edges from neighbors of leaves.  ...  A total acquisition move in a weighted graph G moves all weight from a vertex u to a neighboring vertex v, provided that before this move the weight on v is at least the weight on u.  ...  By definition, every non-leaf vertex of a tree is a core vertex or has a leaf neighbor.  ...

### Power Optimization in Fault-Tolerant Topology Control Algorithms for Wireless Multi-hop Networks

M.T. Hajiaghayi, N. Immorlica, V.S. Mirrokni
2007 IEEE/ACM Transactions on Networking
Specifically, we require all links established by this power setting be symmetric and form a k-vertex connected subgraph of the network graph. This problem is known to be NP-hard.  ...  Our third algorithm assumes that the edge lengths of the network graph form a metric.  ...  In the optimal integral solution, there should be at least one edge incident to each vertex; thus the cost of an optimal integral solution is at least n since p v ≥ 1 for all v.  ...
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