Ramanujan Coverings of Graphs

Let $G$ be a finite connected graph, and let $\rho$ be the spectral radius of its universal cover. For example, if $G$ is $k$-regular then $\rho=2\sqrt{k-1}$. We show that for every $r$, there is an $r$-covering (a.k.a. an $r$-lift) of $G$ where all the new eigenvalues are bounded from above by $\rho$. It follows that a bipartite Ramanujan graph has a Ramanujan $r$-covering for every $r$. This generalizes the $r=2$ case due to Marcus, Spielman and Srivastava (2013). Every $r$-covering of $G$ corresponds to a labeling of the edges of $G$ by elements of the symmetric group $S_{r}$. We generalize this notion to labeling the edges by elements of various groups and present a broader scenario where Ramanujan coverings are guaranteed to exist. In particular, this shows the existence of richer families of bipartite Ramanujan graphs than was known before. Inspired by Marcus-Spielman-Srivastava, a crucial component of our proof is the existence of interlacing families of polynomials for complex reflection groups. The core argument of this component is taken from a recent paper of them (2015). Another important ingredient of our proof is a new generalization of the matching polynomial of a graph. We define the $r$-th matching polynomial of $G$ to be the average matching polynomial of all $r$-coverings of $G$. We show this polynomial shares many properties with the original matching polynomial. For example, it is real rooted with all its roots inside $\left[-\rho,\rho\right]$.Comment: 38 pages, 4 figures, journal version (minor changes from previous arXiv version). Shortened version appeared in STOC 201

On The Total Irregularity Strength of Regular Graphs

Let ðº = (ð‘‰, ð¸) be a graph. A total labeling ð‘“: 𑉠∪ ð¸ → {1, 2, ⋯ , ð‘˜} iscalled a totally irregular total ð‘˜-labeling of ðº if every two distinct vertices ð‘¥ and𑦠in 𑉠satisfy ð‘¤ð‘“(ð‘¥) ≠ ð‘¤ð‘“(ð‘¦) and every two distinct edges ð‘¥1ð‘¥2 and ð‘¦1ð‘¦2 in ð¸satisfy ð‘¤ð‘“(ð‘¥1ð‘¥2) ≠ ð‘¤ð‘“(ð‘¦1ð‘¦2), where ð‘¤ð‘“(ð‘¥) = ð‘“(ð‘¥) + Σð‘¥ð‘§âˆˆð¸(ðº) ð‘“(ð‘¥ð‘§) andð‘¤ð‘“(ð‘¥1ð‘¥2) = ð‘“(ð‘¥1) + ð‘“(ð‘¥1ð‘¥2) + ð‘“(ð‘¥2). The minimum 𑘠for which a graph ðº hasa totally irregular total ð‘˜-labeling is called the total irregularity strength of ðº,denoted by ð‘¡ð‘ (ðº). In this paper, we consider an upper bound on the totalirregularity strength of ð‘š copies of a regular graph. Besides that, we give a dual labeling of a totally irregular total ð‘˜-labeling of a regular graph and we consider the total irregularity strength of ð‘š copies of a path on two vertices, ð‘š copies of a cycle, and ð‘š copies of a prism ð¶ð‘› â–¡ ð‘ƒ2

Computing the Edge Irregularity Strengths of Chain Graphs and the Join of Two Graphs

In computer science, graphs are used in variety of applications directly or indirectly. Especially quantitative labeled graphs have played a vital role in computational linguistics, decision making software tools, coding theory and path determination in networks. For a graph G(V,E) with the vertex set V and the edge set E, a vertex k-labeling $\phi: V \rightarrow \{1,2,\dots, k\}$ is defined to be an edge irregular k-labeling of the graph G if for every two different edges e and f their $w_\phi(e) \ne w_\phi(f)$, where the weight of an edge $e=xy \in E(G)$ is $w_\phi(xy)=\phi(x)+\phi(y)$. The minimum k for which the graph G has an edge irregular k-labeling is called the edge irregularity strength of G, denoted by es(G). In this paper, we determine the edge irregularity strengths of some chain graphs and the join of two graphs. We introduce a conjecture and open problems for researchers for further research

Group Irregularity Strength of Connected Graphs

We investigate the group irregularity strength ($s_g(G)$) of graphs, i.e. the smallest value of $s$ such that taking any Abelian group \gr of order $s$, there exists a function f:E(G)\rightarrow \gr such that the sums of edge labels at every vertex are distinct. We prove that for any connected graph $G$ of order at least 3, $s_g(G)=n$ if $n\neq 4k+2$ and $s_g(G)\leq n+1$ otherwise, except the case of some infinite family of stars

Ideal Basis in Constructions Defined by Directed Graphs

The present article continues the investigation of visible ideal bases in constructions defined using directed graphs. This notion is motivated by its applications for the design of classication systems. Our main theorem establishes that, for every balanced digraph and each idempotent semiring with identity element, the incidence semiring of the digraph has a convenient visible ideal basis. It also shows that the elements of the basis can always be used to generate ideals with the largest possible weight among the weights of all ideals in the incidence semiring

Derandomization Beyond Connectivity: Undirected Laplacian Systems in Nearly Logarithmic Space

We give a deterministic O˜(log n)-space algorithm for approximately solving linear systems given by Laplacians of undirected graphs, and consequently also approximating hitting times, commute times, and escape probabilities for undirected graphs. Previously, such systems were known to be solvable by randomized algorithms using O(log n) space (Doron, Le Gall, and Ta-Shma, 2017) and hence by deterministic algorithms using O(log3/2 n) space (Saks and Zhou, FOCS 1995 and JCSS 1999). Our algorithm combines ideas from time-efficient Laplacian solvers (Spielman and Teng, STOC ‘04; Peng and Spielman, STOC ‘14) with ideas used to show that UNDIRECTED S-T CONNECTIVITY is in deterministic logspace (Reingold, STOC ‘05 and JACM ‘08; Rozenman and Vadhan, RANDOM ‘05).Engineering and Applied Science