Let g(n, l1, l2, d, t, q) be the number of general 4-regular graphs on n labelled vertices with l1 + 2l2 loops, d double edges, t triple edges and q quartet edges. We use inclusion and exclusion with five types of properties to determine the asymptotic behavior of g(n, l1, l2, d, t, q) and hence that of g(2n), the total number of general 4-regular graphs where l1, l2, d, t and q = o( √ n), resp...

Let G be a connected graph of order n. The resistance matrix of G is defined as RG = (rij(G))n×n, where rij(G) is the resistance distance between two vertices i and j in G. Eigenvalues of RG are called R-eigenvalues of G. If all row sums of RG are equal, then G is called resistance-regular. For any connected graph G, we show that RG determines the structure of G up to isomorphism. Moreover, the...

Let G be a connected reductive linear algebraic group over C and let (; V) be a regular representation of G. There is a locally-nite representation (^ ; C V ]) on the aane algebra C V ] of V deened by ^ (g)f(v) = f(g ?1 v) for f 2 C V ]. Since G is reductive, (^ ; C V ]) decomposes as a direct sum of irreducible regular representations of G. The representation (; V) is said to be multiplicity f...

Let J3 be the number of vertices commonly adjacent to any pair of non-adjacent vertices. It is proved that every strongly regular graph with even order and J3 ~ 1 is l-extendable. We also show that every strongly regular graph of degree at least 3 and cyclic edge connecti vity at least 3k -3 is 2-extendab Ie. Strongly regular graphs of k even order and of degree k at least 3 with J3 ~"3 are 2-e...

An edge dominating set F of a graph G is a subset of E(G) such that every edge in E(G) \ F is incident with at least one vertex that is an end-point of an edge in F . Edge dominating sets of small cardinality are of interest. We refer to the size of a smallest edge dominating set of a graph G as the edge domination number of G and denote this by β(G). In this paper we improve all current known ...

the degree kirchhoff index of a connected graph $g$ is defined as‎ ‎the sum of the terms $d_i,d_j,r_{ij}$ over all pairs of vertices‎, ‎where $d_i$ is the‎ ‎degree of the $i$-th vertex‎, ‎and $r_{ij}$ the resistance distance between the $i$-th and‎ ‎$j$-th vertex of $g$‎. ‎bounds for the degree kirchhoff index of the line and para-line‎ ‎graphs are determined‎. ‎the special case of regular grap...

For each nonnegative integer r we determine a set of graph operations such that all r regular loopless graphs can be generated from the smallest r regular loopless graphs by using these operations We also discuss possible extensions of this result to r regular graphs of girth at least g for each xed g

Let G = (V,E), V = {1,2, . . . ,n}, be a simple graph without isolated vertices, with vertex degree sequence d1 ≥ d2 ≥ ·· · ≥ dn > 0, di = d(i). A graph G is regular if and only if d1 = d2 = · · · = dn. A graph invariant I(G) is measure of irregularity of graph G with the property I(G) = 0 if and only ifG is regular, and I(G)> 0 otherwise. In this paper we introduce some new irregularity measures.

Let G be a simple and simply connected complex linear algebraic group, with Lie algebra g. Let ρ : G → AutV be an irreducible finite-dimensional representation of G, and let ρ∗ : g → EndV be the induced representation of g. The goal of this paper is to study ρ∗, and in particular to give normal forms for the action of ρ∗(X) for regular elements X of g. Of course, when X is semisimple, the actio...

When the graph is a path with n vertices, we simply call PG the path polynomial and denote it by Pn. Define An as the adjacency matrix of a path on n vertices. For several interesting classes of graphs, A(Gi) is a polynomial in A(G), where Gi is the ith distance graph of G ([5]). Actually, for distance-regular graphs, A(Gi) is a polynomial in A(G) of degree i, and this property characterizes th...