I refer to the background of the notion of $Linear~complex$ by A. A. Zykov in the article entitled "General properties of Linear Complexes"
A $Linear~complex $ (or simplex complex) is a finite set $L$ for which the set $(L)^2$ of all pairs is split into non- intersection parts: $(L)^2=K+\bar{K,}$ with the following conditions:
(I) Two pairs, which differ only in the order of the elements, belongs to one and the same part.
(II) The pairs of equal elements belongs to $\bar{K}$.
The number $e(L)$ of elements of the complex $L$ is called its order.
The elements $A$ and $B$ of $L$ are called adjacent $(A\omega B)$, if the pair of them belongs to $K$, and not adjacent $(A\bar{\omega} B)$, if it belongs to $\bar{K}.$ The condition I expresses the symmetric property of the sign $\omega$, the condition II its anti- reflexiveness.
The complexes $L_1$ and $L_2$ are equal $(L_1=L_2$ if they consist of the same elements and if any two elements which are adjacent in $L_1$ are adjacent in $L_2$, and conversely.
The complementary complex to $L$ is that complex $\bar{L}$ which consists of the same elements but in which any two distinct elements which are adjacent in $L$ are not adjacent in $\bar{L}$, and conversely.
Now, I wonder as what are the points of difference between a $Linear~ complex $ and a $ simple~ graph$? Or are they both same?
Yes, linear complexes are the same things as graphs. (Simple graphs, if you must, though I think it's more common to say "graph" and "multigraph" than "simple graph" and "graph".) But Zykov is writing to us from an alien universe (namely, the year 1949) which does not yet have a well-developed notion of graphs.
These days, Zykov is considered to be one of the founders of graph theory as a discipline. By 1969, when he was writing a textbook on graph theory, he called it the Theory of Finite Graphs. But how was he supposed to know what to call a graph in 1949?