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The purpose of this article is to improve existing lower bounds on the chromatic number chi. Let mu_1,...,mu_n be the eigenvalues of the adjacency matrix sorted in non-increasing order. First, we prove the lower bound chi >= 1 + max_m {sum_{i=1}^m mu_i / - sum_{i=1}^m mu_{n-i+1}} for m=1,...,n-1. This generalizes the Hoffman lower bound which only involves the maximum and minimum eigenvalues, i.e., the case $m=1$. We provide several examples for which the new bound exceeds the {\sc Hoffman} lower bound. Second, we conjecture the lower bound chi >= 1 + S^+ / S^-, where S^+ and S^- are the sums of the squares of positive and negative eigenvalues, respectively. To corroborate this conjecture, we prove the weaker bound chi >= S^+/S^-. We show that the conjectured lower bound is tight for several families of graphs. We also performed various searches for a counter-example, but none was found. Our proofs rely on a new technique of converting the adjacency matrix into the zero matrix by conjugating with unitary matrices and use majorization of spectra of self-adjoint matrices. We also show that the above bounds are actually lower bounds on the normalized orthogonal rank of a graph, which is always less than or equal to the chromatic number. The normalized orthogonal rank is the minimum dimension making it possible to assign vectors with entries of modulus one to the vertices such that two such vectors are orthogonal if the corresponding vertices are connected. All these bounds are also valid when we replace the adjacency matrix A by W * A where W is an arbitrary self-adjoint matrix and * denotes the Schur product, that is, entrywise product of W and A.

Communication is the main motive in any Networks whether it is Wireless Sensor Network, Ad-Hoc networks, Mobile Networks, Wired Networks, Local Area Network, Metropolitan Area Network, Wireless Area Network etc, hence it must be energy efficient. The main parameters for energy efficient communication are maximizing network lifetime, saving energy at the different nodes, sending the packets in minimum time delay, higher throughput etc. This paper focuses mainly on the energy efficient communication with the help of Adjacency Matrix in the Wireless Sensor Networks. The energy efficient scheduling can be done by putting the idle node in to sleep node so energy at the idle node can be saved. The proposed model in this paper first forms the adjacency matrix and broadcasts the information about the total number of existing nodes with depths to the other nodes in the same cluster from controller node. When every node receives the node information about the other nodes for same cluster they communicate based on the shortest depths and schedules the idle node in to sleep mode for a specific time threshold so energy at the idle nodes can be saved., Comment: 20 pages, 2 figures, 14 tables, 5 equations, International Journal of Computer Networks & Communications (IJCNC),March 2012, Volume 4, No. 2, March 2012

Recently there is huge interest in graph theory and intensive study on computing integer powers of matrices. In this paper, we investigate relationships between one type of graph and well-known Fibonacci sequence. In this content, we consider the adjacency matrix of one type of graph with 2k (k=1,2,...) vertices. It is also known that for any positive integer r, the (i,j)th entry of A^{r} (A is the adjacency matrix of the graph) is just the number of walks from vertex i to vertex j, that use exactly k edges.

Consider any random graph model where potential edges appear independently, with possibly different probabilities, and assume that the minimum expected degree is omega(ln n). We prove that the adjacency matrix and the Laplacian of that random graph are concentrated around the corresponding matrices of the weighted graph whose edge weights are the probabilities in the random model. While this may seem surprising, we will see that this matrix concentration phenomenon is a generalization of known results about the Er\"{o}s-R\'{e}nyi model. In particular, we will argue that matrix concentration is implicit the theory of quasi-random graph properties. We present two main applications of the main result. In bond percolation over a graph G, we show that the Laplacian of the random subgraph is typically very close to the Laplacian of G. As a corollary, we improve upon a bound for the spectral gap due to Chung and Horn that was derived via much more complicated methods. In inhomogeneous random graphs, there are points X_1,...,X_n uniformly distributed on the interval [0,1] and each pair is connected with probability p kappa(X_i,X_j). We show that if \ln n/n<< p<< 1 and kappa is bounded, then the adjacency matrix of the random graph is close to an integral operator defined in terms of kappa. Our main proof tool is a new concentration inequality for matrix martingales that generalizes Freedman's inequality for the standard scalar setting., Comment: 46 pages, submitted

Let $NPO(k)$ be the smallest number $n$ such that the adjacency matrix of any undirected graph with $n$ vertices or more has at least $k$ nonpositive eigenvalues. We show that $NPO(k)$ is well-defined and prove that the values of $NPO(k)$ for $k=1,2,3,4,5$ are $1,3,6,10,16$ respectively. In addition, we prove that for all $k \geq 5$, $R(k,k+1) \ge NPO(k) > T_k$, in which $R(k,k+1)$ is the Ramsey number for $k$ and $k+1$, and $T_k$ is the $k^{th}$ triangular number. This implies new lower bounds for eigenvalues of Laplacian matrices: the $k$-th largest eigenvalue is bounded from below by the $NPO(k)$-th largest degree, which generalizes some prior results., Comment: 23 pages, 12 figures

We define cut-and-paste, a construction which, given a quadriculated disk obtains a disjoint union of quadriculated disks of smaller total area. We provide two examples of the use of this procedure as a recursive step. Tilings of a disk $\Delta$ receive a parity: we construct a perfect or near-perfect matching of tilings of opposite parities. Let $B_\Delta$ be the black-to-white adjacency matrix: we factor $B_\Delta = L\tilde DU$, where $L$ and $U$ are lower and upper triangular matrices, $\tilde D$ is obtained from a larger identity matrix by removing rows and columns and all entries of $L$, $\tilde D$ and $U$ are equal to 0, 1 or -1., Comment: 20 pages, 17 figures

Motivated by its relevance to various types of dynamical behavior of network systems, the maximum eigenvalue $\lambda_Q$ of the adjacency matrix $A$ of a network has been considered, and mean-field-type approximations to $\lambda_Q$ have been developed for different kinds of networks. Here $A$ is defined by $A_{ij} = 1$ ($A_{ij} = 0$) if there is (is not) a directed network link to $i$ from $j$. However, in at least two recent problems involving networks with heterogeneous node properties (percolation on a directed network and the stability of Boolean models of gene networks), an analogous but different eigenvalue problem arises, namely, that of finding the largest eigenvalue $\lambda_Q$ of the matrix $Q$, where $Q_{ij} = q_i A_{ij}$ and the `bias' $q_i$ may be different at each node $i$. (In the previously mentioned percolation and gene network contexts, $q_i$ is a probability and so lies in the range $0 \le q_i \le 1$.) The purposes of this paper are to extend the previous considerations of the maximum eigenvalue $\lambda_A$ of $A$ to $\lambda_Q$, to develop suitable analytic approximations to $\lambda_Q$, and to test these approximations with numerical experiments. In particular, three issues considered are (i) the effect of the correlation (or anticorrelation) between the value of $q_i$ and the number of links to and from node $i$; (ii) the effect of correlation between the properties of two nodes at either end of a network link (`assortativity'); and (iii) the effect of community structure allowing for a situation in which different $q$-values are associated with different communities., Comment: 15 pages, 3 figures

Paper has been withdrawn due to non-compliance with IJCSI terms and conditions., Comment: Paper has been withdrawn due to non-compliance with IJCSI terms and conditions.

Using the spectral distribution associated with the adjacency matrix of graphs, we introduce a new method of calculation of amplitudes of continuous-time quantum walk on some rather important graphs, such as line, cycle graph $C_n$, complete graph $K_n$, graph $G_n$, finite path and some other finite and infinite graphs, where all are connected with orthogonal polynomials such as Hermite, Laguerre, Tchebichef and some other orthogonal polynomials. It is shown that using the spectral distribution, one can obtain the infinite time asymptotic behavior of amplitudes simply by using the method of stationary phase approximation(WKB approximation), where as an example, the method is applied to star, two-dimensional comb lattices, infinite Hermite and Laguerre graphs. Also by using the Gauss quadrature formula one can approximate infinite graphs with finite ones and vice versa, in order to derive large time asymptotic behavior by WKB method. Likewise, using this method, some new graphs are introduced, where their amplitude are proportional to product of amplitudes of some elementary graphs, even though the graphs themselves are not the same as Cartesian product of their elementary graphs. Finally, via calculating mean end to end distance of some infinite graphs at large enough times, it is shown that continuous time quantum walk at different infinite graphs belong to different universality classes which are also different than those of the corresponding classical ones., Comment: 38 pages, 3 figures, regular paper

We show that the adjacency matrix M of the line digraph of a d-regular digraph D on n vertices can be written as M=AB, where the matrix A is the Kronecker product of the all-ones matrix of dimension d with the identity matrix of dimension n and the matrix B is the direct sum of the adjacency matrices of the factors in a dicycle factorization of D., Comment: 5 pages

Given an m x n rectangular mesh, its adjacency matrix A, having only integer entries, may be interpreted as a map between vector spaces over an arbitrary field K. We describe the kernel of A: it is a direct sum of two natural subspaces whose dimensions are equal to $\lceil c/2 \rceil$ and $\lfloor c/2 \rfloor$, where c = gcd (m+1,n+1) - 1. We show that there are bases to both vector spaces, with entries equal to 0, 1 and -1. When K = Z/(2), the kernel elements of these subspaces are described by rectangular tilings of a special kind. As a corollary, we count the number of tilings of a rectangle of integer sides with a specified set of tiles., Comment: 15 pages, 17 figures

Let $\Delta$ be a bicolored quadriculated disk with black-to-white matrix $B_\Delta$. We show how to factor $B_\Delta = L\tilde DU$, where $L$ and $U$ are lower and upper triangular matrices, $\tilde D$ is obtained from a larger identity matrix by removing rows and columns and all entries of $L$, $\tilde D$ and $U$ are equal to 0, 1 or -1., Comment: 12 pages, 9 figures