We find closed form formulas for the Kemeny’s constant and Kirchhoff index cluster G1{G2} of two highly symmetric graphs G1, G2, in terms parameters original graphs. also discuss some necessary conditions a graph to be symmetric.

A subgroup G of automorphisms of a graph X is said to be 2 -arc-transitive if it is vertexand edgebut not arc-transitive. The graph X is said to be 2 -arc-transitive if Aut X is 1 2 -arc-transitive. The interplay of two different concepts, maps and hypermaps on one side and 2 -arc-transitive group actions on graphs on the other, is investigated. The correspondence between regular maps and 2 -ar...

Slovenija Abstract A vertex-transitive graph is said to be 1 2-transitive if its automor-phism group is vertex and edge but not arc-transitive. Some recent results on 1 2-transitive graphs are given, a few open problems proposed , and possible directions in the research of the structure of these graphs discussed. 1 In the beginning Throughout this paper graphs are simple and, unless otherwise s...

A divisible design graph is a graph whose adjacency matrix is the incidence matrix of a divisible design. Divisible design graphs are a natural generalization of (v, k, λ)-graphs, and like (v, k, λ)-graphs they make a link between combinatorial design theory and algebraic graph theory. The study of divisible design graphs benefits from, and contributes to, both parts. Using information of the e...

Consider a matrix with positive diagonal entries, which is similar via a positive diagonal matrix to a symmetric matrix, and whose signed directed graph has the property that if a cycle and its symmetrically placed complement have the same sign, then they are both positive. We provide sufficient conditions so that A be a P–matrix, that is, a matrix whose principal minors are all positive. We fu...

Unitary graphs are arc-transitive graphs with vertices the flags of Hermitian unitals and edges defined by certain elements of the underlying finite fields. They played a significant role in a recent classification of a class of arc-transitive graphs that admit an automorphism group acting imprimitively on the vertices. In this article, we prove that all unitary graphs are connected of diameter...

Let F be an infinite field with characteristic different from two. For a graph G = (V,E) with V = {1, . . . , n}, let S(G;F) be the set of all symmetric n × n matrices A = [ai,j ] over F with ai,j 6= 0, i 6= j if and only if ij ∈ E. We show that if G is the complement of a partial k-tree and m ≥ k + 2, then for all nonsingular symmetric m ×m matrices K over F, there exists an m× n matrix U such...

A plane graph is called symmetric if it is invariant under the reflection across some straight line. We prove a result that expresses the number of perfect matchings of a large class of symmetric graphs in terms of the product of the number of matchings of two subgraphs. When the graph is also centrally symmetric, the two subgraphs are isomorphic and we obtain a counterpart of Jockusch’s squari...

Motivated by denial-of-service network attacks, we introduce the symmetric interdiction model, where both the interdictor and the optimizer are subject to the same constraints of the underlying optimization problem. We give a general framework that relates optimization to symmetric interdiction for a broad class of optimization problems. We then study the symmetric matching interdiction problem...

we investigate buchbaum and eisenbud's construction of the second symmetric power $s_r(x)$ of a chain complex $x$ of modules over a commutative ring $r$. we state and prove a number of results from the folklore of the subject for which we know of no good direct references. we also provide several explicit computations and examples. we use this construction to prove the following ...